Characterization of the Tangential Boundary Layers in the ...majdalani.eng.auburn.edu/publications/pdf/AIAA20064888.pdf · speed of the rigid-body rotation characteristic of the forced

AIAA 2006-4888Characterization of the Tangential BoundaryLayers in the Bidirectional Vortex Thrust A. B. VyasDepartment of Mathematical SciencesUniversity of DelawareNewark, NE 19716 J. MajdalaniAdvanced Theoretical Research CenterUniversity of Tennessee Space InstituteTullahoma, TN 37388

For permission to copy or republish, contact the copyright owner named on the first page. For AIAA-held copyright,write to AIAA, Permissions Department, 1801 Alexander Bell Drive, Suite 500, Reston, VA 20191-4344.

Propulsion Conference and Exhibit 9–12 July 2006

Sacramento, CA

American Institute of Aeronautics and Astronautics

1

Characterization of the Tangential Boundary Layers in the Bidirectional Vortex Thrust Chamber

Anand B. Vyas* University of Delaware, Newark, DE 19716

and

Joseph Majdalani† University of Tennessee Space Institute, Tullahoma, TN 37388

We consider the tangential boundary layers of the bidirectional vortex, specifically, those forming near the core and sidewall of a swirl-driven cyclonic chamber. The analysis is based on the regularized, tangential momentum equation. The latter is rescaled in a manner to capture the forced vortex near the chamber axis and the no slip requirement at the sidewall. After identifying the coordinate transformations needed to resolve the rapid changes in the regions of nonuniformity, two inner expansions are arrived at. These expansions are then matched with the outer, free vortex solution. By combining inner and outer expansions, uniformly valid approximations are obtained for the swirl velocity, vorticity, and pressure. These are shown to be strongly influenced by the vortex Reynolds number, V. This key parameter appears as a ratio of the mean flow Reynolds number and the product of the swirl number and chamber aspect ratio. Based on V, several fundamental features of the bidirectional vortex are quantified. Among them are the thicknesses of the viscous core and sidewall boundary layers; these decrease with V

1/2 and V, respectively. The converse may be said of the maximum swirl velocity which increases with V

1/2. In the same vein, the angular speed of the rigid-body rotation characteristic of the forced vortex is found to be linearly proportional to V. The form of the swirl velocity is reminiscent of Sullivan’s vortex; here, it is based on the aspect ratio of the chamber. The resulting theoretical predictions are found to be in good agreement with PIV measurements and Navier-Stokes simulations.

Nomenclature a = chamber radius

iA = inlet area b = chamber outlet radius l = chamber aspect ratio, /L a p = normalized pressure, 2/( )p Uρ

iQ = inlet volumetric flow rate iQ = normalized volumetric flow rate, 2 1/( )iQ Ua σ −=

Re = injection Reynolds number, / 1/Ua ν ε= r , z = normalized radial or axial coordinates, /r a , /z a S = swirl number, / iab Aπ πβσ= u = normalized velocity ( ru , zu , uθ )/U uθ = normalized swirl/spin/tangential velocity, /u Uθ U = mean inflow velocity, ( , )u a Lθ V = vortex Reynolds number, 1( / ) ( )iQ Re a L lεσ −= *Visiting Assistant Professor, Department of Mathematical Sciences. Member AIAA. †Jack D. Whitfield Professor of High Speed Flows, Department of Mechanical, Aerospace and Biomedical Engineering. Member AIAA. Distribution A: Approved for public release; distribution unlimited.

42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit9 - 12 July 2006, Sacramento, California

AIAA 2006-4888

Copyright © 2006 by Anand B. Vyas and Joseph Majdalani. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


2

Greek β = normalized outlet radius, /b a δ = η − rescaled radius of the viscous core

cδ = normalized core radius, /c aδ wδ = wall tangential boundary layer thickness, /w aδ

ε = perturbation parameter, 1/ /( )Re Uaν= κ = inflow parameter, 1/(2 ) (2 )iQ l lπ πσ −= λ = η − rescaled wall layer thickness, / aλ ν = kinematic viscosity, /μ ρ η = transformed variable, 2rπ ρ = density σ = modified swirl number, 1 /( )iQ S πβ− = Subscripts i = inlet property r = radial component or partial derivative z = axial component or partial derivative θ = azimuthal component or partial derivative

= overbars denote dimensional variables Superscripts c = composite i = inner core w = near sidewall

I. Introduction T has long been recognized that the free vortex assumption used to model the swirl velocity in columnar vortices deteriorates in the vicinity of the core (e.g., Harvey1 and Leibovich2,3). This is owed to the tangential velocity in a

free vortex being inversely proportional to the distance measured from the axis of rotation. A free vortex overpredicts both the swirl velocity and the radial pressure gradients in the vicinity of the centerline. A similar difficulty has been encountered by Long4 in his classic study of the flow toward a rotating sink. Despite the adequacy of inviscid formulations in describing the genesis of unidirectional vortices, they fall short in capturing important phenomena associated with vortex stability and breakdown. The remedy has been in rescaling the governing equations in order to incorporate the rapid variations that evolve near the core. In the absence of three-dimensional asymmetries, disregarding the wall boundary layer has been shown to be a lesser restriction. Near the wall, viscous forces are sufficiently weak that the axial gradient of a flow variable becomes small in comparison to the radial gradient. Near the core, however, the rescaled equations reveal a forced vortex exhibiting the form suggested by Burgers5 or Oseen and Hamel.6 Accordingly, the complete solution must consist of, first, an outer free vortex and, second, an inner forced vortex whose motion resembles solid-body rotation about the chamber axis. The forced vortex is forged by the viscous stresses that intensify near the core. Due to the increasing shear that accompanies this mechanism, the swirl velocity vanishes along the chamber axis. Bidirectional flows in cyclone separators and combustors exhibit analogous core vortex regions (see Lewellen7). The bipolar motion is further exacerbated by the presence of thin Ekman boundary layers at the endwalls where the swirl velocity is expected to peak and then decay.7,8 These layers are often omitted in theoretical analyses due to their small relative sizes and secondary influence on the bulk flow motion. Evidence of forced vortex behavior near the core of cyclonic chambers has been reported in the experimental and theoretical studies of Kelsall,9 Smith,10,11 Reydon and Gauvin,12 Lin and Kwok,13 and Ogawa.14 By combining experiments and numerical simulations, Hoekstra, Derksen and Van den Akker15 and, more recently, Hu et al.16 have independently confirmed the presence of a viscous core using Laser-Doppler Velocimetry (LDV) and computational fluid dynamics (CFD). Different simulation techniques have been utilized including, to name a few: the Reynolds Stress Model (RSM) by Hu et al.,16 Derksen and Van den Akker,17 and Fang, Majdalani and Chiaverini;18 and the Algebraic Stress Model (ASM) by Boysan, Ayers and Swithenbank.19 Along similar lines,

I


3

Murray et al.20 and Rom, Anderson and Chiaverini21 have acquired computational data and Particle Image Velocimetry (PIV) measurements in cylindrical cyclones. These studies have shown that the swirl velocity reaches a maximum at a small distance cδ from the chamber axis. This radial distance is viewed as a representative lengthscale of the core region surrounding the forced vortex. Furthermore, cδ is found to be nearly invariant along the chamber length. Being consistent with conventional boundary layer theory, cδ diminishes with successive increases in the Reynolds number. Along the sidewall, a companion boundary layer wδ exists but whose analysis has received much less scrutiny. As shown by Vatistas, Jawarneh and Hong,22 the presence of cδ and wδ is characteristic of most vortex chambers, including those driven by either unidirectional or bidirectional flow motion. Despite the number of numerical and experimental investigations of cylindrical cyclones, limited attention has been given to the advancement of suitable mathematical models. The reason could be attributed to the overall complexity of the required equations to solve. In a recent theoretical study of the bidirectional flow in a swirl-driven chamber, a simple inviscid solution was introduced by Vyas, Majdalani and Chiaverini.23 This solution could incorporate the axial dependence along the chamber length while retaining rather realistic inlet and outlet boundary conditions. Insofar as the thrust chamber resembled an inverted cyclone, the analytical solution agreed favorably with existing experimental and numerical predictions of cyclonic flows. It also agreed with previous empirical or semi-analytical correlations that either neglected the axial dependence or relied on elemental regression fits. On the downside, the solution lacked one important ingredient: viscosity. In consequence, it could not account for the forced vortex structure forming near the core. The inviscid swirl velocity and radial pressure gradients became unbounded along the centerline, a situation that pled for further improvement. In this study, a boundary layer treatment of both core and sidewall regions is pursued. Our analysis starts by rescaling the momentum equation in the tangential direction. Being the source of singularity, this equation is regularized by retaining the second-order viscous terms. Subsequently, the boundary layer equation is solved using the tools of matched asymptotic expansions. In the act of matching, the free vortex expression is used to represent the outer solution. Our approach permits the construction of a uniformly valid approximation that exhibits the same distinctive features that one observes experimentally or realizes via computation. In light of the ensuing composite solution, the fundamental flow variables are examined. Furthermore, the swirl velocity and pressure are made non-singular and the characters of the forced vortex and sidewall boundary layers are discussed.

II. Mathematical Model Our physical model, nomenclature, normalization, and coordinate system are identical to those adopted by Vyas, Majdalani and Chiaverini (see Fig. 1).23 Note that spatial coordinates and velocities are normalized, respectively, by the radius a and the average wall tangential injection speed U assumed at entry. While the overall boundary conditions remain essentially unchanged, the tangential momentum equation is regularized following standard practice.24 Being axisymmetric, ( )u rθ does not affect the continuity equation. The axial and radial velocity components zu and ru remain as before,23 related via

1 ( ) 0r zru ur r z∂ ∂

+ =∂ ∂

(continuity) (1)

( ) ( ) 0r zu ur z

θ θ∂ Ω ∂ Ω+ =

∂ ∂ (vorticity transport) (2)

r zu uz r θ

∂ ∂− = Ω

∂ ∂ (vorticity) (3)

with boundary conditions

r

2a

L

z

Figure 1. Sketch of the bidirectional vortex chamber and corresponding coordinate system.


4

0

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

(1, ) 0; 2 ( , ) d

z r

r o z i

u r u z

u z Q u r l r r Qβ

π

= =⎧⎪⎨

= = =⎪⎩ ∫ (4)

The basic solution of this set is expressible by

2sin( ) ( )r θ

r u rr θπκ= − +u e e 22 cos( ) zz rπκ π+ e ; 1

2 2 2i iQ Al aL l

κπ π πσ

≡ = = (5)

where 1u rθ−= is obtained for a purely inviscid flow.23 This swirl velocity is deficient by admitting a region of

nonuniformity near 0r = and allowing slip at the sidewall. The singularity at 0r = is characteristic of inviscid swirling flows to the extent of being encountered in both unidirectional and bidirectional motions.25 Its inability to satisfy the no slip at the sidewall is also consistent with the absence of friction. To overcome the core singularity and capture the sidewall boundary layer, one must first revisit the –θ momentum equation and recognize that the deficiency in the inviscid formulation is primarily due to the dismissal of viscous effects. The swirl momentum equation must hence be regularized through retention of second-order viscous terms. After some manipulation, one is left with

( )22

1 1 1 ; rr

ruu u u Uauu Reur r Re Re r r rr

θθ θ θθ ν

∂⎡ ⎤∂ ∂⎛ ⎞+ = = ≡∇ −⎜ ⎟ ⎢ ⎥∂ ∂ ∂⎝ ⎠ ⎣ ⎦ (6)

where Re is the mean flow Reynolds number. In cyclone separators and combustors, Re is of order 510 . Recalling that ( )u u rθ θ= , Eq. (6) reduces to an ordinary differential equation (ODE), specifically,

d d d 1 d 1( ) ( ) ; d d d d

r rr

u u u uu ru rur r r r r r r Reθ θ

θ θε ε⎡ ⎤+ = = ≡⎢ ⎥⎣ ⎦ (7)

Here 510ε −∼ . For simplicity, we let ruθξ ≡ (8) and reduce Eq. (7) into

d 1 d d 0d d d

rur r r r r

ξ ξε ⎛ ⎞ − =⎜ ⎟⎝ ⎠

(9)

Next, we convert the independent coordinate using

2 d d; 2d d

r rr

η π πη

≡ = (10)

This, in turn, transforms Eq. (9) into

2

2d sin d 0

2 ddε ξ η ξκ η ηη

+ = (11)

where 1 2 3/(2 ) (2 ) 10 10iQ l lκ π πσ − − −≡ = −∼ is the inflow parameter. The size of κ causes the axial and radial velocities to remain much smaller than the driver-in-chief, uθ . Equation (11) represents the reduced tangential momentum equation. To set the stage, one may proceed by identifying the endpoint boundary layers that are responsible for the presence of forced vortex motion about the centerline and velocity adherence along the sidewall. But first, we note that the form of the outer solution may be recovered from Eq. (11) by suppressing viscosity. One gets ( )o Cξ = , where (o) denotes an outer approximation and C is a constant.

III. Tangential Corrections

A. Inner, Near Core Approximation In order to bring the swirl velocity to zero along the chamber axis, one must account for the rapid changes that are due to the local emergence of viscous stresses (see Fig. 2). To do so, one may follow Vyas, Majdalani and Chiaverini26 and introduce the slowly varying scale

η

0 π

δ λ

cent

erlin

e

( )

s ηδ ε

= ( )

x π ηλ ε−

=

outer region

Figure 2. Presence of dual endpoint boundary layers affecting the tangential velocity and causing it to vanish both at the centerline (thus giving rise to a forced vortex) and the sidewall (in observance of no slip requirement).


5

( )

s ηδ ε

≡ (12)

This stretching transformation converts Eq. (11) into

2

2 2 2d sin( ) d 0

dd 2s

ss sε ξ δ ξκδ δ

+ = (13)

The resulting expression may be linearized near the core where 0s ≈ . Straightforward expansion of the sine term in Eq. (13) yields

2 2 2

2 22

d 1 d1 ( ) 02 3! dd

s O sss

ε ξ δ ξδκδ

⎡ ⎤+ − + =⎢ ⎥

⎣ ⎦ (14)

Since both diffusive and convective terms must balance near the core, the two members of Eq. (14) will be of the same order when ~ /δ ε κ (15) Having identified the distinguished limit, one may take / 2 lδ ε κ πεσ≡ = . The core boundary layer equation becomes

2 ( ) ( )

2d 1 d 0

2 dd

i i

ssξ ξ

+ = (16)

where the superscript stands for the inner, near-core approximation. Using a series of the form ( ) ( ) ( )

0 1i i iξ ξ δξ= + +… , one may determine successive viscous corrections to any desired order. In this study, a one-

term inner solution of the form ( ) ( )0

i iξ ξ= will be sought. In the interest of simplicity, the subscript denoting the order in δ is discarded hereafter. The requirements that may be associated with Eq. (16) are those due to forced vortex motion and smooth transition to the outer solution, ( )oξ . These conditions translate into

( )

( ) ( )

0, 0, 0

1, ,

i

i o

r s

r s

ξ

ξ ξ

⎧ = = =⎪⎨

→ →∞ =⎪⎩ (17)

Integration of Eq. (16) may be readily carried out with the outcome being

12( )

0 1si C e Cξ −= + (18)

As we insist on forced vortex behavior near the core, we set ( ) (0) 0iξ = and retrieve

12( )

0 ( 1)si C eξ −= − (19) The constant 0C may be determined in harmony with the outer solution. Using Prandtl’s matching principle, we set ( ) ( )

0lim limi o

s ηξ ξ

→ →∞= (20)

and get 0C C= − . A uniformly valid expansion may hence be obtained that does not fulfill the no slip at the wall. This composite inner solution may be determined from

( ) ( ) ( )( )( ) ( ) ( ) ( ) 1 1 2 21 1 12 2 41 exp 1 exp 1 exp

oci i o i C C Cr Vrξ ξ ξ ξ δ η πκε− −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤= + − = = =− − −− − −⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (21)

where V is the vortex Reynolds number,

1 iQRe aVl L Lεσ σ ν

≡ = = (22)

It may be interesting to note the qualitative resemblance between Eq. (22) and the vortex Reynolds number encountered in two cell swirling motion such as Sullivan’s vortex;27 Sullivan’s control parameter is found to be proportional to the flow circulation at infinity and the reciprocal of the kinematic viscosity. At this juncture, the outer constant may be exacted from the downstream condition of a tangentially injected fluid at entry. This implies

( ) (1) 1ciξ = or ( )1

4

11 exp

CV

=− −

(23)

The composite inner solution becomes

21

4

14

( ) 1

1

Vrci

V

e

eξ

−

−

−=

− or

214

14

( ) 1 1

1

Vrci

V

eur e

θ

−

−

⎛ ⎞−= ⎜ ⎟⎜ ⎟−⎝ ⎠ (24)


6

The composite inner approximation is identical to the solution presented by Vyas, Majdalani and Chiaverini.26 Note that as 0ε → , V →∞ , and ( ) 1ciu rθ

−→ ; forthwith, the swirl velocity associated with a free vortex is restored. Conversely, as 0r → at fixed ε , one can expand Eq. (24) into

1 14 4

2 2 41 1( ) 38 96(1 )

( )4(1 ) 4(1 )

ciV V

rV Vr V r rVu O re e

θ − −

− + += = +

− −

… (25)

This expansion unravels the forced vortex relation, ( ) ~ciu rθ ω , where

144(1 )V

V

eω

−−∼ (26)

Here ω represents the angular speed of the core layer which, owing to intense viscous forces, is caused to rotate as a rigid body about the chamber axis.

B. Inner, Sidewall Approximation In similar fashion, the sidewall boundary layer may be captured. Since 0 η π≤ ≤ , one must rescale the thin region near the wall by setting

( )

x π ηλ ε−

≡ (27)

Here λ refers to the thickness of the wall tangential boundary layer. Using (w) to denote a sidewall solution, Eq. (11) may be rearranged into

2 ( ) ( )

2 2d sin( ) d 0

2 ( ) dd

w wxx xx

ε ξ π λ ξλ π λκλ

−− =

− (28)

Subsequently, Eq. (28) may be expanded in the vicinity of the wall to provide

2 ( ) 2 ( )

2 22

d 1 d1 ( ) 02 6 6 dd

w wx O xxx

ε ξ π πλ ξλκλ

⎡ ⎤− − + + =⎢ ⎥

⎣ ⎦ (29)

As before, a distinguished limit of /λ ε κ≡ will ensure comparable orders between diffusive and convective terms. At the outset, Eq. (29) becomes

2 ( ) ( )

22

d 1 1 d1 02 6 dd

w w

xxξ ξπ⎛ ⎞+ − =⎜ ⎟

⎝ ⎠ (30)

Corresponding boundary conditions consist of the no slip at the wall and blending with the composite inner solution in the outer domain. Mathematically, one puts

( )

( ) ( )

1, 0, 0

0, ,

w

w ci

r x

r x

ξ

ξ ξ

⎧ = = =⎪⎨

→ →∞ =⎪⎩ (31)

Forthwith, integration and satisfaction of the no slip at the sidewall lead to ( ) 1 2 2 2 21 1 1 1

0 02 6 4 6exp ( 1)(1 ) 1 exp ( 1) (1 ) 1w K r K V rξ κε π π π−⎡ ⎤ ⎡ ⎤= − − − − = − − − −⎣ ⎦ ⎣ ⎦ (32)

Thereafter, matching with the composite outer solution requires casting

( ) ( )

0 1lim lim 1w ci

r rξ ξ

→ →= = or 21 1

4 60 ( 1)

1

1VK

e π− −=

− (33)

The sidewall approximation becomes

2 21 1

4 6

21 14 6

( 1) (1 )( )

( 1)

1( )1

V rw

V

ere

π

πξ

− − −

− −

−=

− or

2 21 14 6

21 14 6

( 1) (1 )( )

( 1)

1 1

1

V rw

V

eur e

π

θ π

− − −

− −

⎡ ⎤−= ⎢ ⎥⎢ ⎥−⎣ ⎦

(34)

The validity of Eq. (34) is restricted to the region adjacent to the wall. As we move toward the centerline, 0r → , the outer solution is regained. Having identified the core and sidewall boundary layers in a distinctive manner, a unified approximation may be conceived from the triple-merger of outer and inner expansions.

C. Composite, Uniformly Valid Approximation Using Erdélyi’s idea of composite expansions, a uniformly valid solution that extends over the range 0 z l< < may be secured. This solution may be constructed by properly superimposing the endpoint boundary layers while subtracting their common limits. This consolidation yields


7

( )2 22 1 11

4 64

2 21 1 14 4 6

( 1) (1 )( )( ) ( ) ( ) ( ) 2 2 21 1 1

4 4 6( 1)

1 1 1 1 exp exp ( 1) (1 )1 1

V rVroc ci w w

Vr V

e e Vr V re e

π

πξ ξ ξ ξ π

− − −−

− − −

− −⎡ ⎤ ⎡ ⎤= + − = + − − − − − − −⎣ ⎦ ⎣ ⎦− −

(35)

The composite swirl velocity is thus at hand. Note that transcendentally small terms that often arise may be safely ignored due to the prevalent size of the vortex Reynolds number. Recognizing the open injection plane permitted at z l= , one can put

( )

2 22 1 114 64 2 22 1 11

4 641 21 14 4 6

214 21

414

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

( 1)( )

1 1 1 11 1 ; 01 1

1 1 1 1 ; (tangential injection at entry)1

V rVrV rVr

V Vc

VrVr

V

e e e e z lr re e

u ue e z l

r re

ππ

π

θ θ

− − −−− − −−

− − −

−−

−

⎧ ⎡ ⎤− − ⎡ ⎤⎪ ⎢ ⎥+ − − − < <⎢ ⎥⎣ ⎦⎪ ⎢ ⎥− −⎪ ⎣ ⎦= = ⎨⎛ ⎞⎪ −⎜ ⎟ − =⎪ ⎜ ⎟−⎪ ⎝ ⎠⎩

(36)

At the headwall, the presence of an Ekman type boundary layer must cause the swirl velocity to vanish. The treatment of such behavior is deferred to later work.

IV. Results and Discussion

A. Tangential Velocity As shown in Fig. 3, the tangential component of the velocity starts from zero at the wall and then increases rapidly to merge with the outer flowfield within a characteristic distance wδ . It continues to increase until reaching a maximum value that delimits the envelope inside of which viscous forces become dominant. After passing through this maximum max( )uθ , the swirl velocity is seen to gradually depreciate, within a radius cδ , until it reaches zero along the chamber axis. The plot of uθ in Fig. 3 is given at three vortex Reynolds numbers of 210 , 310 , and 410 . Note that increasing the chamber aspect ratio is paramount to magnifying the role of viscosity according to Eq. (22). As shown in Fig. 3a and its inset, the radius of the core vortex cδ expands with successive increases in viscosity. It is largest at the smallest value of V . As the vortex Reynolds number is increased to 410 , the point of maximum swirl draws nearer to the core. This behavior is accompanied by an increase in the magnitude of max( )uθ . With further increases in the vortex Reynolds number, it is clear that uθ approaches the inviscid limit. In the close proximity of the sidewall, Fig. 3b illustrates the rapid damping that the swirl velocity undergoes. The wall tangential boundary layer wδ exhibits a similar dependency on viscosity. Its thickness increases when the vortex Reynolds number is decreased. The rapidly decaying curves appear to be in agreement with experimental measurements acquired by Hu and co-workers.16 They also agree with both CFD and LDV predictions obtained by Hoekstra, Derksen and Van den Akker.15 A two-dimensional vector plot of the swirl velocity is shown in Fig. 4 with and without viscous corrections. Note that the sidewall boundary layer is thinner than the core layer in Fig. 4a. In the interest of validating the shape predicted by the present solution, comparisons are made in Fig. 5 using numerical simulations and experimental measurements obtained, respectively, from Murray et al.,20 and Rom, Anderson and Chiaverini (cf. Fig. 18).21 The numerical data is generated using KIVA, a three-dimensional, finite-volume solver that can handle multiphase, multicomponent, chemically reacting flows. The code is based on a staggered, Arbitrary Lagrangian Eulerian (ALE) technique. The measurements are acquired through Particle Image

0 0.2 0.4 0.6 0.8 10

10

20

30

0 0.1 0.20

10

20

30uθ

r

V = 102

103

104

(uθ )max

δc

core

a) 0.95 0.96 0.97 0.98 0.99 10

5

10

0.990 0.995 10

0.5

1.0

1.5uθ

r

0.99(uθ )outer

δw side

wal

l

b)

Figure 3. Swirl velocity versus /( )iV Q Lν= illustrating the sensitivity of the boundary layer thickness near a) the core and b) the sidewall.


8

Velocimetry (PIV) and an experimental apparatus that is elaborately described by Anderson and co-workers.21,28 In both parts of Fig. 5, numerical and analytical shape predictions show substantial agreement at the estimated vortex Reynolds numbers; furthermore, they are corroborated by actual experimental measurements, especially in the outer and sidewall regions. As the core is approached, the experimental velocity tapers off, falling short of the maximum theoretical values projected by computations and asymptotics. The reduced fidelity of the PIV technique in the vicinity of the forced vortex region is not surprising; it may be attributed to the intensification of viscous drag on seeded particles. Similar trends are depicted in the RSM data and rich LDV measurements taken recently by Hu and co-workers (cf. Fig. 8).16 Their LDV data acquisition system also deteriorates in the core neighborhood. Several arguments could be offered as plausible explanations. While approaching cr δ= from the outer domain, particle drag increases and the swirl velocity becomes quite high relative to the radial velocity; it becomes difficult for the particles to follow the flow or for the experimenter to achieve good seeding concentrations. To increase the precision of measurements, it is essential to use correlated particle pairs from two pulsed laser planes. Thus by using a more elaborate, two-component LDV system, it may be possible in the future to more faithfully capture the high swirling speeds.29

B. Viscous Core Thickness In order to quantify the forced vortex region, it is helpful to select a characteristic lengthscale that is commensurate with the size of the rigid-body, irrotational flow region in which viscous forces are appreciable. For this purpose, we choose the radial distance to max( )uθ as our characteristic length cδ . As shown in the inset of Fig. 3, this distance extends from the chamber axis to the center of the overlap region where the outer and inner core solutions merge. Since the core region is confined to 0 cr δ≤ ≤ , we define the diameter of the forced vortex to be twice this distance. To make headway, we recall that maxc rδ = where maxr is the appropriate zero of the radial derivative of uθ . To obtain this root, one may differentiate Eq. (36) or, equivalently, Eq. (24). We choose the latter and write

viscoussidewall

viscouscore

a) r

uθ

b) r

viscouscore

c) r

inviscid

Figure 4. Qualitative vector plot of the swirl velocity using a) the viscous composite ( )cuθ , b) composite inner ( )ciuθ , and c) inviscid outer ( )ouθ solutions.

0 0.2 0.4 0.6 0.8 10

0.5

1

max( )u

uθ

θ

r

V = 350 CFD20

EXP21

a) 0 0.2 0.4 0.6 0.8 10

0.5

1

b)

r

V = 650 CFD20

EXP21

Figure 5. Analytical swirl velocity ( )cuθ versus computational and experimental predictions from Murray et al.20 and Rom, Anderson and Chiaverini.21 Solutions are at two vortex Reynolds numbers corresponding to a) 350, and b) 650.


9

( )( )

214

14

( ) 2

2

max

2d 2 0d 2 1

Vrci

Vr r

eu Vrr r eθ

−

−

=

−+= =−

(37)

or ( )

21max42

max 2 02 VreVr − − =+ (38)

A root to this transcendental relation may be readily extracted. One finds

( )1 12 21

max 22 1 2 pln 1, / 2.24181/c r e V Vδ −⎡ ⎤= = − − − −⎢ ⎥⎣ ⎦ (39)

where pln( , )x y represents the product log function. At the outset, the diameter of the forced vortex is expressible by 4.48362 /cD V= . Clearly, the thickness of the viscous core is inversely proportional to the square root of the vortex Reynolds number. This result is typical of boundary layers in steady, non-swirling flows. A plot of cδ versus V is given in Fig. 6. Evidently, the curve also represents the locus of the maximum swirl velocity. Its invariance with the axial coordinate may be ascribed to the neglect of thin Ekman-type layers forming at the endwall. Equation (39) permits calculating the maximum swirl velocity for an arbitrary inlet area ratio, chamber aspect ratio, swirl number, and mean flow Re . Through backward substitution into Eq. (24), one gets

( )

( ) ( )

12

11 24

1 12 2

max12

1 exp pln 1,( )

1 2 pln 1,1 V

e Vu

eeθ

−

−−

⎡ ⎤− + − −⎢ ⎥⎣ ⎦=⎡ ⎤− − − −− ⎢ ⎥⎣ ⎦

120.3191V (40)

As shown in Fig. 7, max( )uθ increases with successive increases in the vortex Reynolds number. This is due to the driving speed at entry being (i) entirely tangential, and (ii) directly proportional to the vortex Reynolds number. For the practical range of 2 410 10V≤ ≤ , cδ decreases from 22.42 to 2.24 percent while the maximum velocity is computed to be 3.19 to 31.9 times larger than the average injection speed. The presence of such high tangential velocities is corroborated in past and recent experiments by Anderson and co-workers.21,28

C. Wall Tangential Layer Thickness The size of the sidewall boundary layer may be quantified by defining wδ to be the distance from the wall to the point where the tangential velocity reaches 99% of its final, outer value. To find this point, we set ( ) ( )0.99c ou uθ θ= . Being in the close proximity of the wall, this condition translates into ( ) ( )0.99w ou uθ θ= , where ( )wuθ is given by Eq. (34). Letting wr be the radius at which wall effects become negligible, one solves

( )

214

14

1 exp (1 )1 1111 exp 100

w

w w

V rr rV

αα

⎧ ⎫− ⎡ ⎤− −⎪ ⎪ ⎛ ⎞⎣ ⎦ = −⎨ ⎬ ⎜ ⎟− − ⎝ ⎠⎪ ⎪⎩ ⎭; 21

6 1 0.644934α π≡ − (41)

to get

102 103 1040

0.2

0.4

0.6

0.8

1

0

10

20

30 δc

(uθ )max

(uθ )max

δc

V

Figure 6. Variation of the core layer thickness (left) and maximum tangential speed (right) with respect to the vortex Reynolds number.

101 102 103 10410-3

10-2

10-1

1

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

δw

V

Figure 7. Variation of the wall tangential boundary layer thickness with respect to the vortex Reynolds number. The two-term expression is valid for V > 107.


10

( )1

44 ln 0.01 0.99exp 4ln(100) 28.56211 1 1 1 1 1 1w w

Vr

V V Vα

δα α

⎡ ⎤+ −⎣ ⎦= − = − + − − − − (42)

The latter approximation is valid for 49V > . Furthermore, an expansion of the radical may be attempted to obtain

24 ln10 14.281 7.1405ln10 ln10 11 2 8w V V VV V

δα α α

⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎛ ⎞= + ++ + + ⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎣ ⎦…… (43)

Bearing a relative error of less than 1% for 722V > , a one-term approximation, 14.281/w Vδ , may be utilized for sufficiently large V . The foregoing expressions are compared in Fig. 7 to confirm that the wall tangential boundary layer is inversely proportional to the vortex Reynolds number. This explains the reduced thickness relative the core layer. To further illustrate this point, both wδ and cδ are compared on a log-log scale in Fig. 8. In addition to the clear asymptotic slopes of order one and one-half, respectively, the graph displays the wall-to-core ratio. This ratio decreases by nearly one order of magnitude, from 0.69 to 0.064, as the vortex Reynolds number is increased from

210 to 410 . The wall-to-core ratio is represented by the chained line in Fig. 8. It may be estimated from the asymptotic expressions obtained heretofore. These render

6.3703 7.140528.56210.446068 11 1w

c

VVVV

δδ

⎛ ⎞ ⎛ ⎞+ +− − ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠… ; 49V > (44)

D. Vorticity Correction The region corresponding to 0 cr δ≤ ≤ is the viscous core flanked circumferentially by an outer field that is largely inviscid. Flow rotationality in the outer region is slightly altered due to viscous interactions with the forced vortex. A reassessment of vorticity leads to

2 21 1

2 2 4 41 14 4

exp( ) exp[ (1 )]4 sin( )

2 1 exp( ) 1 exp( )Vr V rVrz r

V Vα α

π κ πα

⎧ ⎫− − −= + −⎨ ⎬− − − −⎩ ⎭

θ zΩ e e (45)

By way of verification, it may be helpful to note that, as 0ε → , the vorticity of the inviscid solution is quickly restored.23 A plot of the axial-to-total vorticity is given in Fig. 9 to illustrate the accelerated decay of the viscous-induced zΩ at larger V . This may be calculated from

2 21 1

4 4 (1 )12

Vr V rz V e e αα− − −⎡ ⎤Ω −⎢ ⎥⎣ ⎦

(46)

These results are shown in Fig. 9a across the chamber radius for / 1z l = and several values of V and σ . In Fig. 9b, we find the dependence on /z l to be marginal except at very low V or when approaching the headwall. In that vicinity, vorticity is everywhere dominated by its axial component. The same occurs near the centerline. Overall, we find that the sidewall boundary layer correction reduces the variability of the vorticity with the axial distance. In reference to Eq. (25), the near-core motion is prescribed by a linear relation between the swirl velocity and the radial coordinate as 0r → . Being of the form u rθ ω= , the angular speed of the forced vortex can be estimated from Eq. (26). Accordingly, so long as 10V > , one can take / 4Vω = . The characteristic angular speed of the rotating core layer is thus at hand.

102 103 10410-3

10-2

10-1

1

10-2

10-1

1

δc

δw

δw/δc

δ

V

Figure 8. Relative sizes of inner core and sidewall boundary layers.


11

E. Pressure Behavior In view of the viscous corrections affecting the tangential velocity, a reassessment of the radial component of the pressure gradient is necessary. Forthwith, the pressure gradient can be re-derived from the classic Euler equation. One finds

2 22

4 64

24 4 6

2( 1)(1 )

3 ( 1)

1 1 1 11 1

VV

V V

rrp e er r e e

π

π

− − −−

− − −

⎡ ⎤∂ − −⎢ ⎥= + −∂ ⎢ ⎥− −⎣ ⎦

2 2 2 2 23

1 sin( ) sin(2 ) 2 cos(2 )r r r rrκ π π π π⎡ ⎤+ −⎣ ⎦

2 22

4 64

2( 1)(1 )3 2 3 2 2 2 21 sin( ) sin(2 ) 2 cos(2 )

VV rrr e e r r r r rπ

κ π π π π− − −−− −⎡ ⎤ ⎡ ⎤− − + −⎣ ⎦⎢ ⎥⎣ ⎦ (47)

The axial pressure gradient is simpler; it collapses into a single expression that is independent of r ,

2 24p zz

π κ∂= −

∂ (48)

Partial integration is straightforward because Eqs. (47) and (48) are independent of z and r , respectively. The spatial distribution of the pressure becomes

( )212144

22 12 2 2 2 2 2 210 22 1 1 1 sin

V rVrp p z r e e rα

π κ κ π⎛ ⎞⎛ ⎞ − −−− ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟⎝ ⎠ ⎝ ⎠

⎡ ⎤= − − − + − − +⎢ ⎥

⎣ ⎦( ) ( )

1 14 41 1 1

4 2 4Ei EiV VVe e V Vα αα α α− −⎡ ⎤− −⎢ ⎥⎣ ⎦

( ) ( ) ( ) ( )1 12 42 2 2 21 1 1 1 1

4 4 2 2 4Ei Ei Ei EiV VV Vr Vr e Vr e Vrα αα α α α− −⎡ ⎤+ − − − + −⎢ ⎥⎣ ⎦ (49)

where Ei( )x refers to the second exponential integral function. It is given by

1

Ei( ) ln!

m

m

xx xm m

γ∞

=

= + +∑ ; 0.5772156649γ (Euler’s constant) (50)

Note that 0p is a reference pressure that depends on operating conditions. Ordinarily, one would consider the suitability of the pressure at the origin, (0,0). However, based on experimental observations and numerical simulations, more confidence in the data is achieved at 1r = .15,16 Near the core, not only is a visible scatter in the data frequently reported but the centerline values seem to be sensitive to the model in question.20,21 In the bidirectional vortex, another factor is relevant. The pressure at the sidewall is the largest due to the role played by centrifugal forces. Near the centerline, a suction effect is created. It is therefore more convenient to take

0 (1,0)p p= , the maximum pressure at the headwall, to be our reference pressure. This choice helps to normalize the pressure data, 0/p p , on an interval [0,1] . By reexamining the relations obtained heretofore, it is apparent that inclusion of viscosity in the tangential momentum balance has an appreciable impact not only on the pressure distribution but, also, on the swirl velocity, mean flow vorticity, and radial pressure gradient. This behavior is illustrated next. The pressure distribution and its radial gradient are displayed at the headwall in Fig. 10. From the graph, one infers that the pressure drop increases in the radial direction due to the effect of centrifugal forces. The pressure distribution in the axial direction is almost negligible when compared to the rapid radial variations. With the advent

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

a)

zΩΩ

104

103

z/l = 1 σ = 1 σ = 10 σ = 100

V = 102

r 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

b)

zΩΩ

104

103

r

σ = 25 z/l = 0.25 z/l = 0.50 z/l = 0.75 z/l = 1.00

102

Figure 9. Axial-to-total vorticity shown a) at the chamber aft end for select values of V and σ , and b) at equispaced chamber locations for select values of V and fixed 25σ = .


12

of viscous corrections, the pressure gradient in Fig. 10a passes through a maximum as one approaches the core. This peak can be calculated from Eq. (47). The pressure peak shown in Fig. 10a increases in magnitude and moves closer to the axis of the chamber when V is increased. This behavior is, of course, consistent with a forced vortex. When the radial pressure gradient is normalized by its inviscid value in Fig. 10b, it becomes virtually independent of l or σ ; instead, it remains a strong function of V . The same can be said of the pressure which is shown in Fig. 11. The actual pressure is normalized by its maximum radial value 0 (1,0)p p= taken at the headwall. Results are shown in Fig. 11 side-by-side with the numerical (KIVA) calculations obtained by Murray and co-workers.20 The case for comparison corresponds to that of Fig. 5a. The simulated runs are plotted at three different times. These show the same trends especially in the outer and wall regions. All numerical curves converge to the same value at the sidewall. However, as the vortex core is approached, deviations in the data start to appear. At the centerline, a minor scatter is observed. The scatter may be attributed to the fidelity of the numerical code in capturing the physics of the flow in the forced vortex region. It may also be connected to the increasing resolution required near 0r = . A strikingly similar pressure behavior is reported by Hu et al.16 and Hoekstra, Derksen and Van den Akker.15 Based on Eq. (47), one can calculate the radial distance pδ corresponding to the point of maximum /p r∂ ∂ . Starting with

max

d 0d r

pr r

∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠ (51)

an asymptotic expression for maxp rδ = can be obtained, specifically,

121.48351/p Vδ (52)

The relative error associated with Eq. (52) is negligible, being less than 0.037 % for 100, 1V σ≥ ≥ and less than 53.5 10−× % for 100, 25V σ≥ ≥ . This error drops precipitously with increasing V or σ . Note that the maximum

0 0.2 0.4 0.6 0.8 11

102

104

0 0.08 0.16101

105pr

∂∂

l = 1–5σ = 1–100

a)

V = 102

V = 103

V = 104

r 0 0.2 0.4 0.6 0.8 10

0.5

1

l = 1–5σ = 1–100

b)

( )

(0)

pr

pr

ε∂∂

∂∂

r

Figure 10. Radial variation at select values of V and l for a) the pressure gradient, and b) the pressure gradient referenced to its inviscid value.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1p0

p

r

V = 350 CFD, t = 0.0675 s CFD, t = 0.0774 s CFD, t = 0.0884 s

Figure 11. Analytical and CFD simulations acquired at the headwall. CFD results are obtained at three different times using KIVA by Murray and co-workers.20

101 102 103 1040

0.2

0.4

0.6

0.8

1

101

102

103

104

( )max/p r∂ ∂

max

pr∂⎛ ⎞

⎜ ⎟∂⎝ ⎠ δp

δp

V

Figure 12. Maximum radial pressure gradient and its locus versus V .


13

radial pressure gradient is closer to the core than the maximum swirl velocity. This is due to / 0.662p cδ δ (53) In view of 1/ 2

p c Vδ δ −∼ ∼ , the thickness of the core is confirmed to be inversely proportional to 1/ 2V (see Fig. 12). Having determined pδ , the corresponding pressure gradient can be evaluated from

3/ 2

3/ 221

max 2

0.0548466 0.0548466[1 exp( )]

p V Vr V∂

=∂ − −

(54)

The relative error in Eq. (54) is insignificant, namely, below 0.137 % for 100, 1V σ≥ ≥ and below 0.069 % for 100, 25V σ≥ ≥ . The maximum radial pressure gradient is plotted on the right-hand-side of Fig. 12.

F. Wall Stress Tensor The shear stresses may be calculated from

2

2 22

sin( )2 2cos( )rrr

u rV rr r

πτ ε ε ππ

∂ ⎡ ⎤= = −⎢ ⎥∂ ⎣ ⎦ (55)

2

22

1 sin( )2 2r ru u u rVr r r r

θθθ

πτ ε ε εθ π

∂⎛ ⎞= + = = −⎜ ⎟∂⎝ ⎠ (56)

2 22 2 cos( )zzz

u V rz

τ ε ε π∂= =

∂ (57)

( ) ( )2 21 1 1

4 4 4

1 14 4

(1 )2 21 12 2

2

2 2 11 coth81 1

Vr V r Vr

r V V

e r V e r V eu uu Vr rr r r r r r e e

α αθ θ

θ α

αετ ε εθ

− − − −

− −

⎡ ⎤+ − − −∂ ∂ ∂⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥= + = = + − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂ ∂ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ − −⎣ ⎦

( ) ( )2 21 14 4 (1 )2 21 1

2 22 2 2 2Vr V re r V e r Vr

αε α− − −⎡ ⎤+ + − −⎣ ⎦ (58)

1 0zz

uur z

θθτ ε

θ∂∂⎛ ⎞= + =⎜ ⎟∂ ∂⎝ ⎠

(59)

2 22 sin( )z r zzr

u u u Vrz rr z r

τ ε ε πε π∂ ∂ ∂⎛ ⎞= + = = −⎜ ⎟∂ ∂ ∂⎝ ⎠ (60)

It is clear that the resultant is dominated by the tangential shear stress, rθτ . At the wall, these quantities become

( ) 2 ( ) ( ) 2 ( ) ( ) ( )122 ; 0; 2 ; ; 0; 0w w w w w w

rr zz r z zrV V Vθθ θ θτ ε τ τ ε τ αε πακ τ τ= = = − = − = − = = (61)

The resultant shear stress is hence ( ) 2 2 21 1 10 2 2 61 32 / ( 1) 2.026w V Vτ αε ε α αε π π κ κ= − + − = − − = − . Contributions

due to ( )wrrτ and ( )w

zzτ are negligible, being of 2( )O ε .

G. Swirling Intensity The swirling intensity defined by Chang and Dhir30 is applied to our problem following the lines described by Vyas, Majdalani and Chiaverini;23 in the context of a bidirectional vortex, one recuperates,

2

1/ 2 1/ 2

0 0

1Ω d d4 z zu r r u u r rθ

−⎛ ⎞= ⎜ ⎟⎝ ⎠∫ ∫ (62)

Using symbolic programming, one may integrate, rearrange and simplify Eq. (62) into

1 14 4erf (1 ) 4 erfi (1 ) 4(1 ) 1 (1 ) (1)

4 2 4 4

i iV i iVi i Cz iV iV

π ππ πκ π π π

⎧ ⎫⎡ ⎤ ⎡ ⎤+ − + +− +⎪ ⎪⎣ ⎦ ⎣ ⎦Ω = − + −⎨ ⎬− +⎪ ⎪⎩ ⎭

(63)

where transcendentally small terms are ignored; here 1i = − , (1) 0.779893C and ( )C x is the Fresnel integral,

2 2 5 4 9 131 1 12 40 34560

( ) cos( )d ( )x

C x r r x x x O xπ π π= = − + +∫ (64)

The product of the inflow parameter and swirling intensity zκ Ω is plotted in Fig. 13 as function of V. The product is shown to vary between 0.2 and 0.838. Note that Ω increases near the headwall and at higher vortex Reynolds numbers; this behavior is particularly favorable to applications that are poised to profit from higher rates of mixing and turbulence. Examples abound and one may cite ORBITEC’s bidirectional vortex engine as one prime beneficiary because of its fuel injection near the headwall.31


14

V. Conclusions In this study, a viscous correction is applied to the bidirectional vortex appropriate of an idealized swirl-driven liquid propellant thrust chamber. The viscous corrections prevent the swirl velocity and pressure from becoming unbounded along the centerline. They also enable us to bring the tangential velocity to zero at the sidewall. Based on the analytical results, the thickness of the forced vortex is characterized as function of the chamber aspect ratio, the swirl number, and the flow Reynolds number. We find the viscous core to decrease with the square root of the vortex Reynolds number, a byproduct of the mean flow Reynolds number, the swirl number, and the chamber aspect ratio. When the kinematic viscosity or the chamber length are increased, the diameter of the forced vortex is magnified proportionately with 1/ 2

c Vδ −∼ . Being proportional to / iL Qν , the core thickness increases when the chamber length or viscosity are increased, and when the injected volumetric flow rate is reduced. It typically varies between 2 and 22 percent of the chamber radius; it may be calculated from 2.2418 /c Vδ as V is reduced from 10,000 to 100. The wall boundary layer wδ is found to be smaller, varying between 0.1 and 15 percent of the chamber radius. Its relative thickness with respect to the core layer diminishes with successive increases in V . It decreases from 0.7 to 0.06. At Reynolds numbers exceeding 700, it becomes directly proportional to / iL Qν and may be approximated from 14.281/w Vδ . The inclusion of viscous forces leads to rapid damping of the swirl velocity; unencumbered, the latter can climb from 3 to as high as 32 times the average injection speed. Our analysis tracks its rise and subsequent fall at both the centerline and sidewall. Similar trends are exhibited by the pressure and its radial gradient. Friction engenders a small non-zero axial vorticity component that vanishes as the sidewall is approached. Due to intense friction near the chamber axis, the viscous core rotates as a solid cylinder with an average angular speed of Vω ∼ . Being proportional to /( )iQ Lν , the angular frequency of the forced vortex increases by increasing the volumetric flow rate. It also increases in shorter chambers with smaller viscosity. Our results help to quantify some of the key features of the bidirectional motion while increasing our repertoire of engineering approximations for confined swirling flows.

Acknowledgments A. B. Vyas is most thankful for the support received from the Department of Mathematical Sciences, University of Delaware. J. Majdalani is deeply grateful for the support received from the National Science Foundation (CMS-0353518) and ORBITEC (FA8650-05-C-2612). The authors appreciate the production of experimental and numerical data by Mark H. Anderson of the University of Wisconsin, Department of Engineering Physics, and Martin J. Chiaverini of Orbital Technologies Corporation (ORBITEC), Madison, WI.

References 1Harvey, J. K., “Some Observations of the Vortex Breakdown Phenomenon,” Journal of Fluid Mechanics, Vol. 14, 1962, pp. 585-592. 2Leibovich, S., “The Structure of Vortex Breakdown,” Annual Review of Fluid Mechanics, Vol. 10, 1978, pp. 221-246. 3Leibovich, S., “Vortex Stability and Breakdown: Survey and Extension,” AIAA Journal, Vol. 22, No. 9, 1984, pp. 1192-1206.

101 102 103 1040

0.2

0.4

0.6

0.8

1zκ Ωzκ Ω

V

Figure 13. Swirling intensity product, zκ Ω .


15

4Long, R. R., “Sources and Sinks at the Axis of a Rotating Liquid,” Quarterly Journal of Mechanics and Applied Mathematics, Vol. 9, 1956, pp. 385-399. 5Burgers, J. M., “A Mathematical Model Illustrating the Theory of Turbulence,” Advances in Applied Mechanics, Vol. 1, 1948, pp. 171-196. 6Schlichting, H., Boundary-Layer Theory, 7th ed., McGraw-Hill, New York, 1979. 7Lewellen, W. S., “A Review of Confined Vortex Flows,” NASA, Technical Rept. CR-1772, July 1971. 8Vatistas, G. H., “Tangential Velocity and Static Pressure Distributions in Vortex Chambers,” AIAA Journal, Vol. 25, No. 8, 1987, pp. 1139-1140. 9Kelsall, D. F., “A Study of Motion of Solid Particles in a Hydraulic Cyclone,” Transactions of the Institution of Chemical Engineers, Vol. 30, 1952, pp. 87-103. 10Smith, J. L., “An Experimental Study of the Vortex in the Cyclone Separator,” Journal of Basic Engineering-Transactions of the ASME, 1962, pp. 602-608. 11Smith, J. L., “An Analysis of the Vortex Flow in the Cyclone Separator,” Journal of Basic Engineering-Transactions of the ASME, 1962, pp. 609-618. 12Reydon, R. F., and Gauvin, W. H., “Theoretical and Experimental Studies of Confined Vortex Flow,” The Canadian Journal of Chemical Engineering, Vol. 59, 1981, pp. 14-23. 13Vatistas, G. H., Lin, S., and Kwok, C. K., “Theoretical and Experimental Studies on Vortex Chamber Flows,” AIAA Journal, Vol. 24, No. 4, 1986, pp. 635-642. 14Ogawa, A., “Estimation of the Collection Efficiencies of the Three Types of the Cyclone Dust Collectors from the Standpoint of the Flow Pattern in the Cylindrical Cyclone Dust Collectors,” Bulletin of the JSME, Vol. 27, No. 223, 1984, pp. 64-69. 15Hoekstra, A. J., Derksen, J. J., and Van den Akker, H. E. A., “An Experimental and Numerical Study of Turbulent Swirling Flow in Gas Cyclones,” Chemical Engineering Science, Vol. 54, 1999, pp. 2055-2065. 16Hu, L. Y., Zhou, L. X., Zhang, J., and Shi, M. X., “Studies of Strongly Swirling Flows in the Full Space of a Volute Cyclone Separator,” AIChE Journal, Vol. 51, No. 3, 2005, pp. 740-749. 17Derksen, J. J., and Van den Akker, H. E. A., “Simulation of Vortex Core Precession in a Reverse-Flow Cyclone,” AIChE Journal, Vol. 46, No. 7, 2000, pp. 1317-1331. 18Fang, D., Majdalani, J., and Chiaverini, M. J., “Simulation of the Cold-Wall Swirl Driven Combustion Chamber,” AIAA Paper 2003-5055, July 2003. 19Boysan, F., Ayers, W. H., and Swithenbank, J., “A Fundamental Mathematical Modelling Approach to Cyclone Design,” Institute of Chemical Engineers, Vol. 60, 1982, pp. 222-230. 20Murray, A. L., Gudgen, A. J., Chiaverini, M. J., Sauer, J. A., and Knuth, W. H., “Numerical Code Development for Simulating Gel Propellant Combustion Processes,” 52nd JANNAF Paper (Technical), May 2004. 21Rom, C. J., Anderson, M. H., and Chiaverini, M. J., “Cold Flow Analysis of a Vortex Chamber Engine for Gelled Propellant Combustor Applications,” AIAA Paper 2004-3359, July 2004. 22Vatistas, G. H., Jawarneh, A. M., and Hong, H., “Flow Characteristics in a Vortex Chamber,” The Canadian Journal of Chemical Engineering, Vol. 83, No. 6, 2005, pp. 425-436. 23Vyas, A. B., Majdalani, J., and Chiaverini, M. J., “The Bidirectional Vortex. Part 1: An Exact Inviscid Solution,” AIAA Paper 2003-5052, July 2003. 24Balachandar, S., Buckmaster, J. D., and Short, M., “The Generation of Axial Vorticity in Solid-Propellant Rocket-Motor Flows,” Journal of Fluid Mechanics, Vol. 429, 2001, pp. 283-305. 25Bloor, M. I. G., and Ingham, D. B., “The Flow in Industrial Cyclones,” Journal of Fluid Mechanics, Vol. 178, 1987, pp. 507-519. 26Vyas, A. B., Majdalani, J., and Chiaverini, M. J., “The Bidirectional Vortex. Part 2: Viscous Core Corrections,” AIAA Paper 2003-5053, July 2003. 27Sullivan, R. D., “A Two-Cell Vortex Solution of the Navier-Stokes Equations,” Journal of the Aerospace Sciences, 1959, pp. 767-768. 28Anderson, M. H., Valenzuela, R., Rom, C. J., Bonazza, R., and Chiaverini, M. J., “Vortex Chamber Flow Field Characterization for Gelled Propellant Combustor Applications,” AIAA Paper 2003-4474, July 2003. 29Anderson, M. H., University of Wisconsin, Madison, Private Communication, 2006. 30Chang, F., and Dhir, V. K., “Mechanisms of Heat Transfer Enhancement and Slow Decay of Swirl in Tubes with Tangential Injection,” International Journal of Heat and Fluid Flow, Vol. 16, No. 2, 1995, pp. 78-87. 31Chiaverini, M. J., Malecki, M. J., Sauer, J. A., Knuth, W. H., and Majdalani, J., “Vortex Thrust Chamber Testing and Analysis for O2-H2 Propulsion Applications,” AIAA Paper 2003-4473, July 2003.

Documents

Characterization of the Tangential Boundary Layers in the ...majdalani.eng.auburn.edu/publications/pdf/AIAA20064888.pdf · speed of the rigid-body rotation characteristic of the forced