Inviscid and viscous models of the vortex breakdown phenomenon

NSF GRANT GK-1297 THE OFFICE OF NAVAL RESEARCH

REPORT NO. AS-67-14 AND THE

AUGUST 1967 NATIONAL SCIENCE FOUNDATION

INVISCID AND VISCOUS MODELS OF THE VORTEX BREAKDOWN PHENOMENON

by

Hartmut H. K. Bossel

Submitted in partial satisfaction of the requirements

for the degree of Doctor of Philosophy in Engineering

Distribution of this report is unlimited

FACULTY INVESTIGATOR: M. Holt, Professor of Aeronautical Sciences

August 1967

University of California Aeronautical Sciences Division

ABSTRACT

In the case of high swirl and large Reynolds number, the

Navier-Stokes equations for rotationally symmetric incompressible flow are shown

to reduce to 1) a viscous parabolic system for slender (quasicylindrical)

flows and 2) an inviscid elliptic system for expanding (or contracting) flows.

The inviscid system is solved for the case of flow with initial rigid rotation in a cylindrical stream surface. Assuming different

down-stream boundary conditions, Fourier-Bessel series solutions are computed

for the supercritical (nonoscillatory) case and plotted. For very high

swirl values, closed and open bubbles of recirculating fluid are obtained

for certain cases, where the closed bubbles resemble those observed in vortex tube experiments.

The viscous slender problem is formulated in an integral method,

using approximating functions for axial velocity and circulation which satisfy boundary and asymptotic requirements. Weighting functions are

used to generate a linearly independent set of equations of sufficient

number to determine the coefficients in the velocity and circulation

approximations as functions of one (the axial) coordinate. The formu-lation is for any order of approximation. Computational difficulties appear near the suspected breakdown point. Flows resembling breakdown flows are obtained.

Page ABSTRACT TABLE OF CONTENTS ii LIST OF FIGURES v NOMENCLATURE vii 1 - INTRODUCTION 1

2 - VORTEX BREAKDOWN OBSERVATIONS 4

A survey of experiments 5

Definition 5

Features of vortex breakdown 6

Occurrence of vortex breakdown 7

Delta wing 7

Confined vortex flows 8

Conclusions 9

3 - THEORIES OF VORTEX BREAKDOWN 10

Separation analogy 10

Hydrodynamic instability 11

Standing wave 12

Finite transition

4 - AN AMPLIFICATION CRITERION 14

5 - EQUATIONS OF MOTION AND BOUNDARY CONDITIONS 17

Equations of motion 17

Two limiting cases 19

Slender vortex flows 19

Expanding vortex flows 20

Transformation of variables 22

Boundary conditions ₂₄

6 - INVISCID SOLUTIONS ₂₇

A general linear solution ₂₇

Swirling flow in a cylindrical streamtube ₃₀

Pressure on the axis ₃₃

Velocities ₃₄

Nondimensional results ₃₅

Specific cases ₃₆

Computational note ₃₉

7 - INVISCID RESULTS AND DISCUSSION ₄₀

Parameters ₄₀

Stream surface plots 41

Velocity on the axis ₄₂

Axial velocity profiles ₄₂

Swirl velocity profiles ₄₃

Swirl angle profiles ₄₄

Summarizing discussion 45

An explanation of breakdown 47

8 - VISCOUS SOLUTIONS ₄₉

Method of solution 49

Integral relations ₅₂

Approximations for axial velocity and circulation 54

Axial velocity W ₅₄

Circulation K ₅₅

Weighting functions ₅₆

The numerical method 61

Singularities 62

Computing programs 62

Flow models ₆₃

Results for zero swirl ₆₇

Results for increasing swirl 67

Summary and discussion of the viscous results 70

REFERENCES ₇₃

FIGURES ₇₉

Inviscid results _{79 - 127} Viscous results 128 - 138

LIST OF FIGURES Inviscid results

Fig. 1 _{Coordinate system and velocities} Fig. 2 _{Vortex breakdown in a tube}

Fig. 3 Vortex core model

Fig. 4 _{Boundary conditions on} ip

Fig. 5 _{Pressure on the axis for cases A and B with}

R0 = 0.4 Figs. 6 _{Stream surfaces, velocity on axis, and axial}

velocity profiles for cases A - F with zero swirl and L = 2

Figs. 7 _{Stream surfaces, velocity on axis, and axial}

velocity, swirl

velocity, and swirl angle profiles for cases A - F with initial swirl

_{v Lw}

= 1.75 and L = 2

c

Figs. 8 _{Stream surfaces, velocity on axis, and axial}

velocity, swirl

velocity, and swirl angle profiles for cases A - F with initial swirl v 7w = 1.91585 and L = 2

c

Figs. 9 _{Stream surfaces, velocity on axis, and axial}

velocity, swirl velocity, and swirl angle profiles for cases A - F with initial swirl

_v/w

= 1.191585 and L = 4

Viscous results

Fig. 10 _{Flow chart for computing program VIVO 1} Fig. 11 _{Flow chart for computing program VIVO 2}

Fig. 12 _{Initial axial and swirl velocity profiles. Ke = 2.0} Fig. 13 _{Velocity on axis, and axial velocity profile development}

for profile B, no swirl

Fig. 14 _{Axial velocity profile development for}

profile C, no swirl Fig. 15 _{Velocity on axis, and axial and swirl velocity}

profile development for profiles A, ke = 2.0

development for profiles A, Ke = 2.8

Fig. 17 Initial axial and swirl velocity profiles, velocity on axis,

and axial and swirl velocity profile development for profiles

A, Ke = 3.2

Fig. 18 Velocity on axis, axial and swirl velocity profile development,

and stream surfaces for profiles B, Ke = 3.6

Fig. 19 Velocity on axis, axial and swirl velocity profile development, and stream surfaces for profiles B, Ke = 4.0

Fig. 20 Velocity on axis, axial and swirl velocity profile development, and stream surfaces for profiles B, Ke = 41

NOMENCLATURE

(Capital variables are nondimensional)

a(Z)

parameters of the velocity profile

b(Z)

parameters of the circulation profile

cik(Z)

coefficients in the system of ordinary differential

d ,k(Z) equations in the method of integral relations

weighting functions

Bessel functions of order zero and one

j1

zeros of the Bessel functions of order one

k=vr _circulation

h =u r

9. reference length

p pressure

p0

= p + --

(u2+v2+w2), total pressure r radial coordinate Re Reynolds number u radial velocity v swirl velocity w axial velocity y

=r2/2

z _{axial coordinate}

a(Z) exponent parameter of the velocity profile (Z) _{exponent parameter of the circulation profile}

y stream surface angle with respect to axial direction

2 V r w

C

p density

= arctan V/W, swirl angle stream function

Subscri pts

ax on axis

c outer edge of rigidly (or nearly rigidly) rotating core

e external flow

i initial

reference

Superscri pts

differentiation with respect to z

- INTRODUCTION

Since its first recorded observation in 1957, vortex breakdown

has been the object of numerous investigations. There is no lack now of experimental observations, and several theories have been advanced to

explain the phenomenon.

Vortex breakdown can occur in the vortex systems of flight

vehicles, seriously affecting their pressure distribution. It may occur in swirl diffusers, influencing their performance. It stabilizes flames and drastically changes the flow from rotating jets. It could be used

to contain a volume of fluid within a body of different fluid with

rela-tively little mixing. The phenomenon may have something to do with the

mechanism of transition to turbulence.

The goal set for this report was to compute vortex breakdown

flows.

Observations of vortex breakdown flows and related flows under

a variety of circumstances are reviewed in Section 2. A definition of vortex breakdown, to be used throughout this report, is based on their

common features. Conclusions about the importance of different flow

parameters are drawn from the experimental results.

The different theories that have been advanced to explain

vor-tex breakdown are briefly outlined in Section 3.

In Section 4 a criterion for stability against an axial velocity

disturbance is derived for a highly simplified model of vortex core flow.

The Navier-Stokes equations for incompressible rotationally

symmetric motion are examined in Section 5. By an order-of-magnitude analysis it is found that inviscid elliptic equations apply in the

(quasi-cylindrical) vortex flow. Boundary conditions and some expected features

of breakdown flows are discussed.

In Section 6 conditions are derived under which the inviscid

equation of motion becomes linear. It turns out that the linear equation

belongs to flows having a rigid rotation at some point along their axis,

a condition which also applies approximately over the inner core of a

viscous vortex.

Solutions are found by the method of separation of variables,

and are specialized to swirling flow in a cylindrical stream tube. The

solutions considered are in the supercritical (nonoscillatory) regime,

characterized by a maximum swirl angle = 62.5. Stream function and velocities are given in a Fourier-Bessel series solution. The series coefficients are obtained for six different cases.

Numerical results for the six cases of Section 6 are obtained

and plotted in Section 7. Closed and open bubbles are obtained for very high swirl values in some cases, bulges without free stagnation points

in others. Vortex breakdown is explained as a feature of flows with very

high swirl values having a retardation of axial velocity near the axis.

Section 8 presents an integral method to solve the viscous

slender system. The system is integrated in one direction by the use of

exponentially decaying approximating and weighting functions. The

func-tional dependence of the parameters in the approximating function is then

determined from ordinary linear first order differential equations. Some

features of the method appear to be novel and could also be applied in

Section 9 discusses results obtained with the integral method

and some computational difficulties. At initial swirl values approaching the critical one, flows resembling breakdown flows are obtained.

2 - VORTEX BREAKDOWN OBSERVATIONS

A survey of experiments

In investigations on swept wings (Peckham and Atkinson, 1957;

Lambourne and Pusey, 1958) the cores of the leading edge vortices (marked

by smoke or condensation) were observed to ubell out' (Pecicham and Atkinson, 1957) at some distance downstream from the apex of the wing. These first

observations spawned a number of experimental and theoretical investigations

into the nature of the phenomenon, now referred to as vortex bursting, or, preferably, vortex breakdown, since it is not a time-dependent phenomenon.

The majority of experiments were carried out on delta wings,

where the modification of force and moment coefficients is of immediate

practical interest (Elle, 1960; Lawford and Beauchamp, 1961; Hummel, 1965).

Vortex breakdown was intensively studied by means of smoke or dye filaments

in wind and water tunnels at very low to moderate speeds (Elle, 1958;

Werlë, 1960; Maltby and Keating, 1960; Das, 1961; Lambourne and Bryer, 1961;

Lowson, 1964; Poisson-Quinton and Werl, 1967). A few observations were

made at sonic and supersonic speeds (Elle, 1960; Lambourne and Bryer, 1961;

Craven and Alexander, l96). Rolls, Koenig, and Drinkwater, 1965, report

on flight observations of vortex breakdown in a humid atmosphere. An abrupt

change in radial vorticity distribution in the trailing vortex of a straight

wing was observed by investigators of the Pennsylvania State University in

1965.

It was soon realized that the phenomenon might be of a more

funda-mental nature and would not only appear in leading edge vortices. Breakdown

was shown to occur in vortex tubes (Lambourne and Bryer, 1961; Harvey, 1962

Irani, 1965). These flows clearly showed a stagnation point and reversed

flow downstream of it.

Stagnation points and reversed flows in swirling fluids have

also been observed in the vortex whistle (Vonnegut, 1954; Chanaud, 1965),

in water flows through a convergent-divergent nozzle and in a tube (Binnie,

Hookings, and Kamel, 1957; Binnie, 1957), in flames in vortex flow (Potter,

Wong, Berlad, 1958), in vortex chambers (Nissan and Bresan, 1961;

Donaldson and Snedeker, 1962), in flows through expanding cross sections

and diffusers (Strscheletzky, 1957; Gore and Ranz, 1964; So, 1967). There

is evidence for a stagnation point and reversed flow in some tornado funnels

(Morton, 1966).

Vortex breakdown has been discussed in survey articles by KUchemann

(1965) and Hall (l966b).

Although there are debatable differences between the flows

men-tioned, their common features will be used to define vortex breakdown for

the purpose of this report.

Definition

Vortex breakdown is the rapid expansion over a short axial

dis-tance (of the order of the thickness of the rotational core) of streamtubes

centered about the axis of a rotationally symmetric vortex whose velocity

components are continuous on the axis.

The breakdown may or may not result in stagnation on the axis

and reversed flow, and it may or may not be accompanied by circumferen-tially periodic flow (uspirallinghi) or breakdown into turbulent flow. The flow may be incompressible or compressible,inviscid or viscous.

This definition covers more flows than have previously been

classified by the term "vortex breakdown."

Features of vortex breakdown

The flows mentioned have in common the features of the defini-tion just given. In some cases (supersonic vortex breakdown, for example)

it is not clear whether a stagnation point and reversed flow exist. In most other cases stagnation points on the axis and axial flow reversal have been observed. In these cases vortex breakdown would appear as

follows (see, for example, Lambourne and Bryer, 1961; Harvey, 1962 [Figure

2]; Hummel, 1965).

The axial filament of fluid is suddenly decelerated to stagnation. The flow deflects as if it were encountering a solid object. The

deflec-tion can be axisymmetric, with the fluid in the axial filament expanding

in an initially egg-shaped surface of revolution which may or may not close

downstream. This is the case in most low-speed vortex tube observations (Harvey, 1962; So, 1967). The deflection can also be asymmetric, with the axial filament deflecting with a kink into a spiral configuration about

the axis. This appears to be the more common case in delta wing flows. The spiral persists for a few turns and then breaks down to turbulence. Downstream of the stagnation point there exists reversed axial flow.

Transition to turbulence may or may not occur at the rear of the

egg-shaped bubble in the case of symmetrical breakdown. Harvey (1962)

obtained a closed bubble downstream of which the initial conditions were

apparently restored, with a second breakdown further downstream. His experiments proved reversibility of the basic phenomenon.

Hummel's 1965 measurements of velocities and pressures upstream

and downstream of breakdown show that axial and swirl velocities, which

are initially 2 - 3 times the free stream velocity, are reduced to very

small positive, or negative, or zero values near the axis after breakdown.

Occurrence of vortex breakdown

Delta wins. On a delta wing at sufficient incidence vortex sheets separate from the leading edge and roll up into conical vortices with origin at the

apex of the wing. The generation of vorticity ends at the trailing edge,

and the vortices continue as cylindrical vortices, under viscous diffusion

of vorticity.

As the incidence of the wing is increased, breakdown first occurs far downstream, moving upstream with increasing angle of attack.

Higher effective leading edge sweep, i.e., decreasing angle

between leading edge and free stream direction due to the geometric combi-nation of sweep, incidence, and yaw shifts the first occurrence of breakdown to higher angles of attack (Elle, 1958). Werlé (1960) demonstrated that breakdown occurs further downstream on a thicker wing, and on a wing with

larger leading edge radius. He also showed that on a very slender cone, a rectangular, and an elliptical wing, breakdown does not occur in the immediate neighborhood of the body.

Elle (1958) found that the breakdown position exhibits hysteresis

with respect to yaw, and Lowson (1964) established hysteresis with respect

to incidence. He found two steady flow states to exist, one with breakdown on the wing and one without. Lowson also noted that the position at which

core instability first occurred could be very different from the final state

The effects of upstream and downstream conditions on vortex

breakdown were investigated in particular by Werl (1960) and Lambourne and Bryer (1961). No significant effect of upstream flow disturbances could be

established. Objects in the path of the laminar vortex (upstream of break-down) could precipitate breakdown, or cause an already established breakdown

to move further upstream. Hummel (1965) used a plate normal to the vortex, and some distance downstream of a delta wing to fix the position of

break-down on the wing, which was then almost insensitive to the insertion of a

probe. Wer1 _{(1960) and Lambourne and Bryer (1961) showed that suction on} the vortex axis could retard or even completely eliminate breakdown. A

jet of fluid emitted in the direction of the flow caused the breakdown to

appear further downstream and the vortex axis to curve towards the jet. The influence of Reynolds number on the burst position is appar-ently small. Werlé (1960) reports some upstream movement with increasing Reynolds numbers while Elle (1960) notes that observations at Re l0 and up to Re = 5 x 106 are in perfect quantitative agreement.

Elle (1960) reports on Schlieren observations of the vortex break-down on delta wings at M = 0.7 to 1.03. _{At these Mach numbers he found} the breakdown position to be approximately the same as for low speeds.

Craven and Alexander (1963) found vortex breakdown on a delta wing at M = 2

to exist at incidences only slightly less than those at which breakdown

occurs at low speeds.

Confined vortex flows: As the swirl is increased in a vortex tube of cylin-drical or expanding cross-section (Harvey, 1962; Kirkpatrick, 1965; _{So, 1967)}

or in rotating flow issuing through an expanding cross section or as a free

fluid with a stagnation point at its upstream end appears on the axis

(Figure 2). As the swirl is increased the bubble proceeds upstream until

(in the cases of Harvey, 1962; Kirkpatrick, 1965; So, 1967) a two-celled

vortex with reversed axial flow is established in the duct.

Gore and Ranz (1964) find that the critical swirl ratio for

breakdown is independent of Reynolds number in the range 20 < Re < 60 x

Concl usi ons

From these experiments the following conclusions can be drawn: Swirl is the primary variable. Breakdown will not occur at low swirl values.

The axial pressure gradient affects the breakdown position. An adverse

pressure gradient can apparently precipitate a breakdown which would

otherwise not occur.

Reynolds number effects are not important to the basic phenomenon. Mach number effects are minor.

Influence of upstream turbulence is small, but influence of downstream

disturbances is pronounced since upstream decay of disturbance is much

3 - THEORIES OF VORTEX BREAKDOWN

Several explanations of vortex breakdown have been proposed. They are variants of one of the four propositions:

Vortex breakdown is a separation phenomenon.

Vortex breakdown is the result of hydrodynamic instability. Vortex breakdown is a standing wave phenomenon.

Vortex breakdown marks a finite transition between two conjugate

flow states, analogous to the hydraulic jump.

Separation

Stagnation on the axis with reversed flow downstream implies a

separation-like phenomenon. Strscheletzky (1957) distinguishes between two

kinds of separation: 1) separation from a wall, caused by viscosity, and

2) the formation of discontinuities under the action of inertia forces, and

depending on the hydrodynamic equilibrium of the flow. Reversed flow regions in swirl diffusers are separated from the basic flow by discontinu-ity surfaces of the 2nd order (discontinuities only in the normal derivatives

of velocity), and his calculations indicate a strong dependence of the

geometry of the back flow region on the swirl distribution in the basic flow.

Talbot (1954) applied a one parameter momentum integral method in

his study of swirling laminar flow in a pipe. For large values of this parameter reversed axial flow results. Using the full viscous system of equations Vaisey (1956) computed swirling flows in a tube by a relaxation

method. Depending on Reynolds number and swirl she finds three regimes of

flow: 1) reversed flow in an annular region between axis and wall; 2) down-stream flow everywhere; 3) reversed flow in a region containing the axis.

Fraenkel (1956) finds reversed flow in an analysis of inviscid pipe flow.

Gartshore (1962, 1963) uses the Karman-Pohlhausen technique and

boundary-layer type approximations to compute the flow of (a) a viscous vortex in a uniform stream parallel to the vortex axis (1962) and (b) a viscous vortex imbedded in a concentric irrotational vortex (1963). The

occurrence of singularities for some choices of the parameters is suggested

to be the result of a violation of the boundary layer type assumptions,

and to indicate the occurrence of asymmetries or large axial gradients.

He proposes that vortex breakdown may correspond to these singularities

in a way analogous to the occurrence of separation in two-dimensional boundary layer flow.

Hall's computations of vortex flows by a finite difference method

(Hall, 1966a) and their failure near observed breakdown positions also lead

him to view the breakdown as a failure of the boundary layer type

("quasi-cylindrical") approximation in analogy to problems in the calculation of two-dimensional boundary layer separation.

Hydrodyriamic instability

Jones (1960) assumes axisymmetric disturbances to the inviscid

flow and looks for conditions for amplification. He does not obtain practi-cal results.

Ludwieg (1960, 1961) derives a stability criterion involving the radial gradients of the axial and swirl velocities _{for helical flow}

between two concentric cylinders by assuming spiral disturbances _to

the inviscid flow. _{The theory is supported by a large number of} experiments (Ludwieg, 1964). _{The criterion is applied to the vortex}

breakdown (Ludwieg 1962, 1965), giving good agreement with experimental

data obtained by Hummel (1965).

Standing wave

Squire (1960) looks for conditions under which a sinusoidal

cylin-drical wave of unspecified wave length could first be sustained. He

con-siders three different swirl distributions and finds critical values of the

ratio swirl velocity/axial velocity of the order of unity. This is in agreement with experimental observations.

Finite transition

Benjamin (1962, 1965, 1967; Fraenkel, 1967) considers inviscid

swirling flow. _{He shows that for cylindrical flow the equation of motion}

in general does not possess a unique solution for the stream function _p

between given endpoints. An indefinitely large number of solutions is obtained by varying the two arbitrary constants in the general solution.

Only two solutions, however, appear to be relevant; their solution curves

p(r) do not intersect except at the endpoints (r = 0, r _{= re).} Hence

a different cylindrical flow is possible at another station of the same

duct, keeping the same distribution of Bernoulli constant and circulation

over their common stream surfaces. The two flows A and B are dynami-cally and kinematidynami-cally conjugate. With the aid of the calculus of vari-ations Benjamin shows

If flow A is supercritical (i.e., cannot support standing waves), then flow B _{is subcritical (standing waves can form).}

The supercritical flow has a smaller value of the momentum flux

r

M = 2

_{f (p}

w2 + p)r dr.

Benjamin then explains vortex breakdown as follows: The flow

changes from a supercritical state to a conjugate subcritical state with

an increase in momentum flux M. If tM is small, it can be made up by the positive wave resistance of a stationary wave train. If M is large, the leading wave breaks, and vigorous turbulence results with a

4 - AN AMPLIFICATION CRITERION

Before going into the analysis of vortex core flows it may be useful

to look at a simplified model.

Consider the inner portion of a physical vortex (Fig. 3). Due to

the action of viscosity it will be in nearly solid body rotation close to

the axis, and its axial velocity is therefore assumed constant (w ₌

_w)

over the radius for 0 < r <

_r.

_{Shear is then nonexistent, and the fluid} is treated as inviscid. _{Let the axial velocity of the cylinder of fluid} change by an amount Aw. The results are derived by neglecting small quan-tities of order A2

By continuity the radius of the cylinder will change and

rc2wc = (r + ir)2(w + tw)

This leads to

AW

-

-(4.1)

where Ar/Aw is the expansion of the stream surface p(r) due to the

disturbance Aw.

By conservation of angular momentum we have for an element of fluid at _rc

rv

= _{(r Ar)(v} + Av)

giving

V

Ar _rc (4.2)

or

Av vc

AW 2w

This is the change of swirl velocity v at _{r = r} due to the axial velocity disturbance w.

The centrifugal balance at r = rc + r is, to first order

p(Vc + Av) (w + Aw)Kw = giving 2 Kw Vc

-

2w

The initial velocity disturbance Aw _{will be amplified, i.e.,} > 1 for AW 2 vc 2wc2 > 1

--Ap

V2AW

p

2w-Aw

c (4.3) (4.4)

If the initial disturbance is decelerating, then further _deceleration

and eventual stagnation will result for

Ar _r + Ar

Replacing v and r by using expressions (4.1) and (4.2) leads to 2

- 2Wc_AW

The change in pressure will result in a new w = Kw by way of the axial momentum balance

or a swirl angle

= arctan > arctan V' 54.8°

wc (4.5)

In a viscous vortex the swirl angle increases from zero at the axis

to a maximum near the maximum for the swirl velocity, after which it again

falls off to zero. We can therefore expect that the swirl angle at this point, commonly taken as the characteristic radius of the viscous core, is of critical importance to the behavior of the vortex flow. In fact,

all observations of breakdown have maximum swirl angles close to the value

given in (4.5) (see, for example, Harvey, 1962; Hummel, 1965). It also

agrees well with the theoretical values given by Squire (1960) and by

5 - EQUATIONS OF MOTION AND BOUNDARY CONDITIONS

Equations of motion

The dimensional equations of motion for viscous incompressible

flow with rotational symmetry are in cylindrical coordinates

r Br Bz = 0 (5.1) 2 =

].a+

B2u u 1 Br Bz r

pBr

2 Br Bz r =

)[-_+i+--i_ v

1 r 2 Br Bz r Br Bz 2 r =

_]2+

B2w

1BW+B2W]

Br Bz p Bz

2rBr

Br

It can be expected that this set of equations is governed by a

number of nondimensional parameters, and that it will reduce to simpler

sets of equations for certain choices of these parameters. To find the

parameters and investigate limiting forms of the set of equations, the

dependent and independent variables are non-diniensionalized. Reference

quantities are chosen such that the dependent variables and their

deriv-atives are of order unity for typical problems with continuous velocities

on the axis. We use

w = _{the external (free stream) axial velocity of the potential}

flow outside of the rotational core at a reference station. vc = the maximum swirl velocity at a reference station. In a

viscous vortex it occurs near the outer edge of the

rota-tional core, and roughly marks the point of transition from

a (nearly) rigid body rotation in the center to (nearly) potential vortex flow outside.

set

r = a radius representing core thickness. In a viscous vortex it is convenient to take _rc where v ₌

v.

= stream surface angle at r ₌

_r.

We have = tan y.

The dimensionless variables are

_r

U . p

--

U =

___

r

wtan-

2 p Vc z V V

-

re/tan

I Vc W w w 00

Substitution into system (5.1) results in the nondimensional

-R R Z -0 (5.2) 2 V

22

1 (

c)

-

_BZtany

2 V

2P(

1

=

-ta -

i

)

wrc

tan y

lU

22U

u

-a-3R2 = ( V

\(

1

[V+iaV+(

any) --

22V

V

-Z R

wr

'

tany

Z R

uw

_-

CP

V 1

wr

tan + i + (tan )2 W I R z

The three parameters of the problem are

wr

a core Reynolds number C

V vC

a swirl parameter

-w

a core stream surface angle y

Two limiting cases

Depending on choice of core Reynolds number, swirl parameter, and

core stream surface angle, system (5.2) will reduce to various simpler sets. Two cases are of special interest to the problem at hand: Slender (nearly parallel) vortex flows and expanding vortex flows, both with swirl parameters

near unity (i.e., near the breakdown condition) and relatively large core

Reynolds numbers (core Reynolds numbers for typical breakdown cases are of

the order of io).

Slender vortex flows. Let

= 0(1)

tan _y << 1

wr

_cC

tan y = 0(1)

Neglecting terms of O(tan2y), the set (5.2) reduces to

U + R R Z -R U + W aV UV - V 1 1 V V

ZR -

wr

tany

R2 0

U + W v 2 _DP + V tan y 1 DW -

-(--)

_wr

DR2 The corresponding dimensional system is

r Dr Dz = _{0 (5.3)} r

pDr

=

V[_+]i V

Dr Dz r Dr2

rDrr21

Br Bz

pBz

Dr

rDr

The same set of equations, although found by different approaches,

was used by Gartshore (1962,1963), and Hall (1965,1966a), who calls it

'quasi-cylindrical

Expanding vortex flows. Let

= 0(1) tan _y = 0(1)

wr

>> V Neglecting terms of 0(

wrc

), the set (5.2) becomes

U + BU + DW R DR BZ - 0 2 v 2 2 2 v

2BP

1

)(_c

V DR BZ

tany

w =

tanyH

UV = 0

u+ww

vc 2P

The corresponding dimensional system is

= 0 r r z + u V U r

u+w+

= 0 z r

-pz

The system is inviscid, showing that in the case of expanding

(or contracting) viscous vortex flows inertia and pressure terms dominate the motion.

The last three equations can be written in the vector form of

the Euler equations

ix

curl J = i-grad p0 where p r p0 =

p+

-2 equation. u =

Introduction of a stream-function p eliminates the continuity

In cylindrical coordinates

w

r r

(5.4)

By Kelvin's theorem the circulation is constant along a stream surface, i.e.,

Similarly

p0 =

p0(p)

Using these concepts, the differential equation for the stream

function in steady rotating inviscid flow can be derived (see, for instance,

Fraenkel, 1956). B2 =

-k---i-2dp

r 0 Br2 r Br Bz2 dtp

pdip

Transformation of variables

The equations of motion in cylindrical coordinates can be simpli-fied by using the new variables

2

y=-

-h = ur

k = yr

The inviscid equation (5.5) then becomes

+2y4

-d (5.6)

The viscous slender system (5.3) transforms to

Bw + By Bz

h--

+ Bk -BW + Bw w- -= 0 (5.7) 32k 2

y

-By 3w = p Bz (5.5)

The equations are nondimensionalized by using the reference length

rc and a reference velocity w00. Then

wr

V Re R =

r

Y = rc

z

=-r VR R U =

.0

H =

wr

h w 00 v = _K e w

wr

k 00 dK dP 2

K2Y2

The system of equations for viscous slender steady rotationally

symmetric rotating flow is

(5.8) (5.9) w w w 00

-

p

w2

200

w r

In the inviscid case there is, of course, no Reynolds number stretching, and the same nondimensional variables are used formally with

Re = 1.

The nondimensional inviscid equation of steady, rotationally

symmetric rotating flow is then: P

3H 3Y + 3H 3Y + 3W 3W = 0 (5.10) K2 4Y2 3K 3K -

2Y1-+ W5- -3Y 3W 3W + 2

_(y.

)

H--

₊

W-7

-In "divergence form," using the continuity equation:

= 0 (5.11) K2 4y2 .- (HK - 2Y - + 2K) +

-h-

(WK) = 0

-(HW-2Y--) -(W2+P)

= 0 Boundary conditions

Viscous axisymmetric vortex flows which have no sources or sinks on the axis require

u = 0 v = 0 , - = 0

Only inviscid flows which satisfy the same conditions on the axis will be

considered in this report.

Symmetry demands, besides (3w/3r) = 0, (3v/3r) = const (z) and (32v/3r2) = 0 _{on the axis, i.e., rigid body rotation and shear-free flow} in an infinitesimal neighborhood of the axis of the viscous vortex. In most vortex flows the departure from rigid body rotation becomes appreciable

only when the radius becomes of the order of the viscous core radius _r

defined earlier. The vortex breakdown bubble observed by Harvey (1962) has a radius of the order of the rigidly rotating part of the

vortex,indi-cating that the upstream boundary conditions for the part of the flow

experiencing the breakdown were nearly those of rigid body rotation. As the flow stagnates on the axis (u = v = w = 0) it follows from the equations of motion that

0

at the stagnation point.

The stream surface extending from the stagnation point is (in

the viscous case: initially) a surface of zero swirl velocity. The swirl

velocity distribution in a continuous solution could therefore, downstream

of the stagnation point have the forms (a) or (b).

N

/

()

'I

(h)

r

An examination of the equation of motion (5.1) in the circumfer-ential direction under the conditions v = 0,

(v/r) = 0,

(2v/r2)J

>>

excludes possibility (b). The conclusion is that unless the bubble surface is a surface of discontinuity (which is likely in some cases,

see Strscheletzky, 1957) the rotation inside the bubble will be opposite

V

to that before stagnation. Reversed rotation has indeed been observed by Hummel (1965) and researchers of the University of Pennsylvania (1965).

The axial velocity near the axis must, of course, be reversed

downstream of the stagnation point. _{The flow close to the inside of the}

bubble surface must later again be in the downstream direction _{to satisfy}

continuity requirements. From the experiments the bubble may be closed or open downstream.

Outer boundary conditions are mentioned with the respective

solutions. _{In the case of an unconfined vortex the flow will be that of} a potential vortex far outside the viscous rotational _core.

6 - INVISCID SOLUTIONS

A general linear solution

The results of the discussions of conditions in a real vortex

and of its equations of motion as it expands support the idea that something

about the mechanism of vortex breakdown may be learned from considering the

inviscid equation (5.5) or (5.6). We shall look for solutions which are linear (to make superposition of solutions possible), and which satisfy the

boundary conditions on the axis:

u=O

,

v=O

,

The equation of steady inviscid rotating motion, Eq. (5.6), is

-

- k+

dp0 d1p _p

This partial differential equation will be linear if the highest

power of i on the right hand side is one. This permits a2 2 p

a0+a1i4+p

k2 b2 2 b 1 Substituting in (6.1) + 2y + 2 ip(b2-2a2y) = 2ya1-b1 An immediate solution is = WOQY

Substitution in (6.2) leads to the conditions

(6.1)

and

giving

p

2a2wQ,y2 = 0 hence a2 = 0

2a 2a1

-b2w,y = - 2a1y hence b2 = , or w

-b1 = 0 _(6.3) 0 = a + a1iP = a0 + a1w,y 2 b = b 2 ₀

and, by the boundary condition on v on the axis, (demanding b0 = 0)

k2 2 2

= a1w00y = yv

Therefore v = v'a1wJ = const r = r

While =

wy gives

w = = w

The solution is therefore a rigid rotation. Equation (6.2) now reduces to

y rc w r

By superposition assume the solution

WJ+1J)1

(6.4)

Substitution in (6.4) leads to the separable partial differential _equation for ₁₁₎₁

v 2

1+2y

2

h

= w C V 2 2 4 ) and assuming Introducing A -r Co = F(y)G(z)

the equation separates into two ordinary differential equations _(i.i is the separation constant)

G" + (x2 - 2)G = 0 2

F" +F =

0 with the solutions

2

G(z) = A e Z + B

e2

Z

and

F(y) = C /7 J1(p v'2) + D _{/57 Y1(1i} /2i)

The general solution of Eq. (6.4) now becomes

22

(y,z) = wj+[Cv J1(p)+D

_{J1()](A e}

-X Z + B e -x Z)

In terms of r = w

(r,z) -

_{+[rJ1(pr)+rY1(r)](}

e22

Z +

e2

Z)

_(6.5)

This solution was given by Long (1953) and was used by Long (1956) to compute the flow in a rotating cylinder into a point sink on the axis.

Fraenkel (1956) solved Eq. (6.1) by application of Fourier

trans-forms to compute rotating flow through contractions and expansions of a

pipe, and past bodies in a pipe. In the case of an expansion he finds reversed flow far downstream near the axis.

An important observation about solution (6.5) is that it becomes

oscillatory when the argument of the square root in the exponentials changes

sign. At this point standing waves can first appear.

Equation (6.1) has a counterpart in stratified flows with

solu-tions analogous to (6.5) (see Yih, 1965, chapter 6).

Swirling flow in a cylindrical streamtube

Solution (6.5) of the equation for inviscid rotating flow (6.1)

will now be specialized to describe the swirling flow within a cylindrical

streamtube which is in solid body rotation at some axial position z. It can be expected from the preceding discussions that this solution will show

some of the features of bounded or unbounded (free or confined) vortex core

flows.

We limit the present solution to the nonoscillatory mode, requiring

> 0

This results in cosh, sinh type solutions. The oscillatory mode would be

periodic and undamped in the z-direction. Both features are not dominant

in breakdown flows, but they will be briefly discussed later.

The problem is elliptic and requires conditions on a closed

bound-ary surrounding the field where the solution is being sought. The following conditions will be prescribed (Fig. 4):

1) (o,z) = 0 w r2 2) p(r,o) = ____

wr2

3) _p(r,z) = 2 4) p(r,i) =

Here

The solution will be constructed by superimposing 2

wr

po

and an infinite series constructed from the basic solution

/22

/22

Vp

-x

z

-Vp

-x

z

e + e ₎

0 on the axis requires D and

if

= 0 in solution (6.5).

Proof:

urn lim cos

r J(r) - J(r)

Y1(r) =

Y(r)

=

vl

_sin For r << 1 Yn(r) (n-i)! 2 n

()

1 2 Y1(r) _- ( .j ) Hence

rY1(pr)

r 2 = 2 ₌

_finite0

_{for r-0.}

Therefore

DrYi(pnr) = 0 for r - 0 requires 0 = 0. 2

The requirement (r,o) _{and the condition of nonoscillatory}

behavior demand a sinh type solution in z.

The condition _(rc,z) c _{on the outer boundary requires}

vanishing of the Bessel function component at _{r = rc} or

Hence

l.Inrc =

un

must be the zeros of the Bessel function and

where Un

The argument of the square root in the exponentials then becomes

.2

v 2 Un2_ 2 = 12 r w c C 2v 2 =

L[ 2(c)]

rc2 31n " w

where _Vc/W is the swirl parameter based on the axial velocity and the swirl velocity at the outer radius at the station where the flow is in

solid body rotation.

It is evident now that the flow will change its mode and switch to oscillatory behavior when

w > ill = 3.83170597...

This corresponds to a swirl angle at radius _r of in

3.8317... = arctan

2 62.5°

Under the stated conditions the solution to the problem becomes

2

wr

(r,z) + r

CJ1(j1

)sinh(Jj - (2 (6.7) rc n= 1 R f(r)r

i31n

) dr = 2 r

sinh [j

- ( )2 9. ] c (6.6) (6.8)

Vc _3.8317...

- 2

C

= 1.9158...

a standing wave of infinite wave length appears. The wavelength shrinks

with increasing swirl parameter, until a second wave of infinite wave length

is superimposed on the flow when = j12. We can therefore expect mixed solutions of the type

w r2 k sin V 2v 2 r . c .2 z 31n (r,z)

= -+ r

_CnJl(ln

i n=1

1/2

2v / r (

C)2Z

+r

E C n=k+l fl1 in ) sinh _{Vain -} ;;-r

for swirl values exceeding = 1.9158... . These solutions may app]y

to breakdown flows where oscillatory behavior of the core filament has been

observed ahead of the breakdown. This will not be further discussed in this report.

Pressure on the axis

The pressure on the axis follows from the Bernoulli equation

p0 = p

+(u2

+

v2 +w2)

with 2

wr

f(r) =

[(r)

2 (6.9)

given by prescribing the downstream flow _iI(r). In a following section

several solutions will be given for different assumptions for

A remark is in order concerning oscillatory flows. As the swirl

With 1

w-and

u=0

on the axis V

=0

we have (p0-p) 1/2 j 2 2v )2 ax 1 2 = w r sinh -r

p12w

c

n1

_c (6.10) where the are given by Eq. (6.8).

The pressure distribution is uniquely determined by the down-stream conditions by way of the common

C.

The problem can then be turned around in a physically more meaningful way:

If the pressure on the axis is prescribed and belongs to the

set given by (6.10) the downstream flow for a given swirl follows uniquely

from Eqs. (6.7) to (6.9).

Velocities

The axial velocity at any point in the region under consideration

follows from w = r ar 2v 2' w + l c n=1 ln n o 31n ) sinh - ( -s- z j

CJ(

r

wr

n w c c (6.11)

rv is a constant on = const, hence using the initial (rigid

or

and

where

Fn* = V =

and the initial stream function distribution

2 = we get rv Vc

-

rc rc Wo:, 2v v c 1 = w r r co Nondimensional results

Introducing the nondimensional variables (5.8) (with the formal substitution Re = 1) if' R

G(Z)

= wrc2 / 2 r rc 1

wr3

w V W w v w

Oax

z = ax -

p/2w2

* .2 2v 2' F sinh n Z 2v

-'--)2L

vJln w rc f(r)r i3'in i:;;- ) dr L -r C (6.12) (6.13) (6.14) (6.15)

Then stream function

(R,z) = R2 + 4R

n1

i1n

R)G(Z)

Pressure on the axis

Ip

)l/2 = 1 + 2

nl

31n

G(Z)

ax Axial velocity W = 1 + 2

in oin

R)G(Z)

n= 1 Swirl velocity 2v V =

c'i'

Bubble surface (R(i=O)): given implicitly by

R2 + 4R J1(j1

R)G(Z)

= 0 n= 1

provided there exists a solution for R 0

Stagnation point: for W = 0 on the axis Z must satisfy

n=l

G(Z) +

_{= 0}

Specific cases

In typical vortex breakdown observations, the phenomenon has been

characteristically linked with one or more of the following features down-stream of the breakdown:

Reversed flow on the axis An object at or near the axis

(6.16)

(6.17)

(6.18)

Unreversed flow on the axis (in the case of a closed bubble)

Cases A through F were designed to simulate these downstream

boundary conditions.

Case A simulates the case of a disk of radius R0 on the axis at Z = L. The outflow has--just like the inflow--uniform axial velocity

W over the cross sections, i.e., the highest power of R in the expression for is two.

In case B there is a slight inflow from downstream on and near the

axis--and corresponding outflow in the downstream direction--inside a surface

0 which passes through R = R0 at Z = L and which divides fluid originating from upstream from that originating downstream. The highest power in the stream function is now four for reasons of symmetry, giving a parabolic distribution of the axial velocity W at Z = L.

In case C a disk of radius R = R0 is simulated on the axis at Z = L. Related to case B, the highest power of R in _L(R) is again four, with a parabolic distribution of the outgoing axial velocity W.

Case D has again a parabolic axial velocity distribution at Z = L

with the highest power of R in L(R) again four. The flow is now assumed to be in the downstream direction everywhere at Z = L, having a velocity on the axis of _{W = Wax}

In case E the highest power of R in _L(R) is six, giving an

axial velocity variation with the fourth power of R at Z = L. Slight

inflow is assumed from downstream on and near the axis, returning to downstream

inside of a stream surface P = 0, which passes through R = R0 at Z = L. In case F, as in case D, there is outflow everywhere at Z = L

with a velocity W _Wax on the axis, and varying as the fourth power of R. Correspondingly the highest power of R in

In the following equations R0 is a radius 0 < R < 1 at

which the stream function becomes zero (i.e., a streamline splitting from

the axial stream line will pass here).

Case A: Downstream obstacle, quadratic relation for

For O<R<R0:

= 0 R2-R 2 For R < R < 1: = l-R0 * R0 F = -2 2 Ji(uinRo) ln (l-R0 )

Case B: Downstream inflow, quartic relation for ''

2R2-R2

L

For 0<R<l:

= R l-R02 *

_4oin

F=

_n (l-R02)j1

Case C: Downstream obstacle, quartic relation for

For 0 < R < R0: _'1( = For R0 < R < 1: L2 = R2 R2-R02 l-R02 (6. 20) (6.21) (6.22) Fn* =

_j(iR02)

R )+R { 31n 2 in 0 0 31n R0 }

Case D: Downstream outflow, quartic relation for

For 0 < R 1: = _R2[W + (1_Wax)R2] * F = --- (l-W ) J (j ) (6.23) n .3 ax 0 in 3ln

Case E: Downstream inflow, sixth power relation for

44

2 R For 0 < R < 1: _'PL(R) = R

lR04

(12

_-

J0(j1)

*

_un

F=

(l-R04) 31n

Case F: Downstream outflow, sixth power relation for

For 0 < R < 1: _{= R2[W}

)R]

ax

96

(12 _{--- )l_Wax)Jo(jin)}

*

_31n

F=

n .3 3m Computatioa1 note

To avoid computational difficulties, replace

sinh(aZ)

sinh(aL)

by ean(Z_L)

e_an+t

1 where a

\Ij

2v 2 =

Cases A through E are computed by using the first 19 terms of the

infinite series expressions.

(6.24)

7 - INVISCID RESULTS AND DISCUSSION Parameters

The parameters of the solutions for cases A through F are

the swirl v _1w

the length ratio i/re = L the radius R0

the axial velocity _Wax at Z = L

Solutions were obtained for different combinations of these

parameters, and computer plots were prepared for the following cases

The swirl value of = 1.91585 is just barely below the theoretical value of j11/2 = 3.8317.. ./2 which divides nonoscillatory

(supercritical) from oscillatory (subcritical) flows. All cases discussed

here belong to the nonoscillatory class.

The results of the calculations are plotted in Figures 6 through

9. They show stream surfaces 'F _{= const, the velocity on the axis, and}

radial profiles at different Z-stations of axial velocity W, swirl velocity V, and swirl angle . Before more general conclusions are

drawn the five different kinds of plots will be discussed separately. L = 2, _Vc/W = 0: R0 = 0.1, Wax = 0.2 (Figs. 6) L = 2,

_v/w

= 1.75: R0 = 0.1, Wax = 0.2 (Figs. 7)

L = 2, _Vc/W = 1.91585: R0 = 0.0,0.1,0.3, Wax = 0.2 (Figs. 8) L = 4, _Vc/W = 1.91585: R0 = 0.1, Wax = 0.2 (Figs. 9)

Stream surface plots

The stream surface plots for the case of no swirl (Figs. 6) have

the features of other nonrotating flows: Substantial upstream influence is

confined to the immediate neighborhood of the disturbance, represented by

the downstream condition on 1'. The picture changes little as swirl is

intro-duced and increased. The character of the flow remains essentially non-rotational up to very high swirl values of the order of 1.75 (or 60°) (Fig. 7). At this value a bubble has appeared in some cases. It grows

rapidly in size and upstream dimension as the swirl is further increased

to the limit value of 1.9185 or q 62.5° (Figs. 8 and 9). At the same

time the upstream influence of the downstream flow increases drastically,

as evidenced by the slope of the stream surfaces.

The free bubbles have the characteristic shape of an egg: a blunt

front, then gradually increasing diameter, finally a fairly rapid decrease

in diameter to a blunt end. These features are also evident in Harvey's (1962) photographs (Fig. 2). The forward portions of the other bubbles, with the exception of case A have the same features. In the case of the greater length parameter (L = 4) the bubbles are more carrot-shaped. They

remind one (especially in cases where L is even greater) of the tips of

certain tornado funnels with stagnation points on their axes (Morton, 1966).

It is important to note that the appearance of the bubbles is a

continuous phenomenon. This becomes expecially clear from case F: At zero

and small swirl this case has simply a velocity retardation on and near the

axis with a corresponding widening of the stream surfaces. The velocity

minimum appears at the end of the region at Z = L. As the swirl is increased a bulge forms in the surface contours, indicating the minimum in the axial

velocity has moved upstream. At some higher swirl value this minimum reaches the value zero on the axis, corresponding to a free stagnation

point, or a hlbubbleu of infinitesimal size. The bubble grows rapidly with

further increase in swirl.

Velocity on the axis

Since U and V are zero on the axis,

P

ax ax

In addition to the plots of velocity on the axis (Figs. 6 - 9), 'ax

as a function of swirl is given in Figs. 5 for cases A and B with R0 = 0.4.

Again there is a striking effect of swirl on the pressure and

velocity distributions. While the changes are slight as swirl is increased

to a value near 1.5 ( 56.3°) they become quite pronounced as it is

increased further to 1.91585. The high swirl permits a much greater upstream

penetration of downstream velocity and pressure disturbances. The effect is less pronounced in case A.

Some cases have velocity minima at some Z < L, corresponding to the bulges in the stream surfaces. Where stagnation points appear the velocities on the axis are reversed and comparatively small downstream of

the first free stagnation point.

In cases B through F the velocity on the axis decays practically

linearly to the stagnation point at high swirl values.

Axial velocity profiles

In the flow models considered the axial velocity at and near the

lower than the constant axial velocity of the solid body rotation at the

upstream end (Figs. 6 - 9). At low--and even fairly high--swirl values the velocity changes monotonically from one to the other distribution.

At very high swirl values the axial velocity near the axis is

retarded even more before it speeds up again to meet the required condi-tions at the downstream end. In some cases this leads to free stagnation points and bubbles with reversed flow near the axis. The axial velocities

in the bubble are of the order of 10% of the initial rigid core velocity,

with the highest values occurring on the axis and on the bubble surface.

Extending from the forward stagnation point there is a surface of zero

axial velocity inside the bubble.

With the exception of the profiles at the downstream end of case

A all velocity profiles are strikingly similar, especially in the bubble

region. The profiles at the breakdown point (first free stagnation point)

are virtually indistinguishable from each other for cases B through F.

Swirl velocity profiles

The initially linear dependence of the swirl velocity on R changes gradually and monotonically to the assumed downstream conditions

when the swirl is not too high (Figs. 6 - 9).

At high swirl values V/R becomes zero at a free stagnation point

on the axis. There are reversed swirl velocities in the bubble with magni-tudes of the order of up to 10% of the initial maximum swirl velocity. On the bubble surface the swirl velocity is zero. For all cases with the

exception of case A, the swirl velocity profiles through the bubble region

and therefore

Since q' = const on the outer boundary _{R = 1, the swirl}

velocity is also constant there.

Swirl angle profiles

In the cases considered the maximum swirl angle always appears on the outer boundary (Figs. 6 - 9). Initially it is given by the swirl

parameter of the problem, i.e.,

qi =

arctan (-i)

Since the swirl velocity on the outer stream surface (R = 1) remains constant while W(R=1) increases, the swirl angle there decreases

accordi ngly.

With increasing Z the swirl angle profiles become increasingly more linear in their outer portions. In cases B through F the breakdown

point is marked by a swirl angle profile which is linear all the way to the

axis. The swirl angle is zero on the bubble surface and then increases,

having again a linear dependence in the outer portion. In the linear region

q/R

has almost the same value at all stations. In this region constant swirl angles are found on nearly conical surfaces.

The linear R-dependence of the swirl at the breakdown point relates

/R at the breakdown to the swirl angle at the boundary at this station.

It turns out that, at the breakdown position, vc

vc /(R=l)

= 2w

For the case of maximum swirl the outer swirl angle at the break-down point is q 54.8°, and

q/R

0.958.

At all other positions

4/R

is higher. If this minimum value of p/3R is not reached on the axis, no stagnation point will appear (see

case 0).

In case F at high swirl rigid body rotation is again restored near

the axis at the downstream end, changing to a flow with constant swirl angle

in the outer portion.

Sumarizing discussion

The computer experiments have shown that under certain conditions

supercritical inviscid flows with initial solid body rotation can have free

stagnation points on the axis. From the first of these a bubble-shaped stream surface originates which contains reversed axial flow on and near the axis and reversed swirl velocities throughout.

If a bubble with stagnation points and reversed flow does not appear, the stream surfaces may still expand and again contract to form a bulge. The

transition from the bulge to the bubble is a smooth one and is just a matter

of sufficient velocity retardation on the axis. Both phenomena are therefore denoted with the common term tvortex breakdown."

*

These relations appear plausible also from Eqs. (6.16), (6.18) and (6.19),

but no rigorous proof could be found.

The phenomenon is critically dependent on the initial swirl of

the flow and o,i the downstream flow conditions. Generally very high swirl

values close to the theoretical maximum for nonoscillatory rotating flows

are needed to obtain a breakdown bubble.

The breakdown flows obtained fall into two groups. In case A

the bubble is flat, does not extend far upstream, and its maximum diameter

is never greater than the "disk" radius R0. Cases B through F, though

having a variety of different downstream boundary conditions, have almost

identical flows in the breakdown region, and their bubbles have similar

shapes. Where free bubbles exist for L = 2 they resemble those observed by Harvey (1962).

The similarity of cases B through F is explained by considering

the expressions for the coefficients Fn* _{in the series solution (Eqs.} 6.20 - 6.25). The dominant terms in the expressions for F for these cases are all of the order while for case A Fn* is of order

_[J1(i1)/i].

Thus a different type of flow is to be expected.

Another type of flow is again to be expected when an eighth (or

higher) power relation in R is assumed for 1. For q' - R8 this would

lead to coefficients Ffl of the order and

This case has not been explored further since a variation of the axial

velocity with the sith power of R appeared as too drastic an assumption.

Given a flow with high swirl and initial rigid rotation, an

essential condition for the appearance of breakdown is then a retardation

of the axial velocity, or equivalently, an increase in the pressure on the

(By the properties of the solution one implies the other.) Breakdown may then appear at a considerable distance ahead of the disturbance.

Any one of the stream surfaces could be considered a boundary

for a different inviscid flow problem, say the flow in a swirl diffuser.

On the inner surfaces the pressure increases, while on the outermost

sur-faces it actually decreases due to the increased axial velocity, and an

almost constant swirl velocity.

The size and upstream extent of the bubble or bulge is dependent

on both the initial swirl and the shape of the downstream "bucket" in the

axial velocity. A dominance of higher powers of R in the expression for

the downstream stream function enlarges the bubble radially and in the

upstream direction. The effect of increasing swirl is similar.

In case F an almost rigid rotation is restored over the inner portion of the flow. _{Cases are conceivable where this region is even} wider, with a swirl value high enough to cause a second breakdown.

An explanation of breakdown

The results indicate that vortex breakdown is _{a phenomenon}

peculiar to supercritical rotating flows which have swirl values very close to the critical swirl value. _{In the case of initial rigid rotation the} critical swirl value is j11/2 _{(1/2 of the first zero of the Bessel} func-tion corresponding to 62.5°). Squire (1960) has given critical values for several different vortex flows.

The necessary conditions for breakdown are: a swirl value close

to the critical one, and a retardation of the axial velocity near the axis at some downstream station.

The phenomenon is a smooth reversible transition of the flow,

and it may or may not lead to stagnation and reversed flow on the axis. In the case of no stagnation a bulge may form in the stream surfaces.

Bubbles may be closed or open downstream and can be of the "egg" or

"carrot" variety.

Where stagnation occurs, the phenomenon can be viewed as a

centrifugal separation in the sense of Strscheletzky (1957).

Spiralling and transition to turbulence are secondary effects which can modify the appearance of the flow.

8 - VISCOUS SOLUTIONS Method of solution

In the preceding two sections we considered special solutions to the inviscid elliptic system (5.4):

= 0 r 3r az = z r

p3r

= 0 3z r =

_i._p

p3Z

In section 5 the Navier-Stokes equations were shown to reduce to

this system when the stream surface angle of a rotationally symmetric vortex

flow with large swirl and large Reynolds number became appreciable, i.e., when tan y = 0(1). This condition definitely applies in the immediate

break-down neighborhood, as can be seen from Harvey's photographs (Fig. 2) and

from the plots of the inviscid solutions.

In the next two sections, solutions will be sought to the viscous parabolic system (5.3) = 0 r r 3z 2 r

pr

= 3z r 2

r3r

2 r + + 1 3w

-- p3Z

v[--This limiting form of the Navier-Stokes equations _{was shown to}

apply in regicns of negligible stream surface angle of rotationally

symetric vortex flows with large swirl and large Reynolds number.

The parabolic nature of this system permits _{a step-by-step}

integration by "marching" downstream, beginning with given initial _values. This is a vast simplification over the (generally nonlinear) elliptic

boundary value problem, and it permits the calculation of vortex flows by methods analogous to those used in boundary layer theory. In this section,

an integral method for the computation of slender vortex flows will be

developed.

It is tempting to try to compute breakdown flows by _{using the} parabolic system, especially since the elliptic (breakdown) region is usually of small extent, and the flow immediately _{outside of it remains} parabolic in nature. _{Attempts to "march" downstream with a finite} dif-ference scheme have nevertheless failed (Hall, 1966a). Gartshore _(1962,1963) encountered difficulties with a Karman-Pohlhausen type integral method. Difficulties must be expected in the neighborhood of breakdown, stemming

from the neglect of the first two (convection) terms in the _{second equation}

of the Navier-Stokes system (5.1) or the inviscid _{system (5.4) which make} those systems elliptic. _{In the breakdown region, they have the same order} of magnitude as the remaining terms of the centrifugal _{pressure balance in} the viscous parabolic system (5.3).

The inviscid solutions have shown that upstream influence

in-creases drastically with swirl, becoming infinite and undamped when the

inviscid flow changes from the supercritical to the subcritical _state.

expected that more severe problems will be encountered in swirling flows

when computation with the parabolic system is attempted.

The best hope of achieving meaningful results despite locally

singular behavior lies with integral methods, and recent applications of

integral methods have had some remarkable success in the computation of

separated flows where a similar difficulty exists at the separation point. (Lees and Reeves, 1964; Nielsen, Lynes, Goodwin, 1966; Holt, 1966.)

The viscous slender vortex problem was therefore formulated in

an integral method which is based on work by Ritz, Galerkin, Kantorovich, Dorodnitsyri and others (for a historical outline, see Bethel, 1966). A

description of the principle of the method can be found in Belotserkovskiy

and Chushkin (1962). For the solution of a system of partial differential

equations in two coordinates the method would consist in approximating the

dependent variables by members of a closed set of functions in one inde-pendent variable, say y, and depending on n parameters

a(z)

which

are functions of the second independent variable, say z. The system of

equations is multiplied by members of a linearly independent set of

weight-ing functions (the set must be closed) to generate a number n of linearly

independent equations equal to that of the unknown parameters, and inte-grated in the y-direction. This generates n ordinary differential

equa-tions which are solved for the n unknown parameters a1(z).

The method will be written in general form for an unspecified

order of approximation N. It will then be applied to vortex flows with

varying degrees of swirl, and specifically to flows where breakdown is to

parabolic system, partly from the mathematical formulation which leaves

room for future improvements.

Integral relations

The divergence form of the equations of motion for slender viscous incompressible rotationally symmetric vortex flow (Eq. 5.11) is

3W

+

(wK) + (HK-2Y + 2K) = 0

---

(W+P)

+ --- (HW-2Y ) = 0

The third and fourth equations are coupled via the second one. To reduce the number of equations, P is eliminated by cross

differenti-ation. The resulting system is

3W +

-

= 0 (8.1) - (WK) + -- (HK-2y + 2K) = 0 (8.2) 3W K2 2 +

----(Hw_2Yfi)

= 0 (8.3) 3Y

In the last two equations H is now considered known from the first

equa-tion by

Y

H =

_-

_f

dY

0

assuming no sources or sinks on the axis.

=0

2 K 4y2

Integral relations are now obtained by multiplying Eq. (8.3) by

a weighting function Eq. (8.2) by a weighting function

g(Y).

and integrating with respect to Y from 0 to .

The integrated equations are

d 3W

[()2

-

f

f(Y)W2dY]

_{+ dZ $} 2 K2dY + (Hw-2Y

00

o4Y

- [f(Y)(Hw-2Y .

_)]+

f

_{f(Y)HWdY-2[f(Y)YW]}

00

0 + 2 'k (Y)Y + f(Y)]WdY = 0 (8.4) +

- f

g(Y)HKdY

_{- 2[gk(Y)Y} 3K

Y]

0

+ 2[g(Y)YK] - 2

f

_{[g(Y)Y+2g(Y)]KdY + 2[g(Y)K]}

= 0 (8.5)

0 0 0

With the assumption that weighting functions and

g(Y)

can be found which will reduce the integrated terms to zero at the limit

of integration, the two equations reduce to

-f

f(Y)W2dY

+ dY

+ f

fj(Y)HWdY + 2

f

[fj(Y)Y

4Y2 0 0

+ f(Y)]WdY

= 0 (8.6) - 2 ₅

_{K[2g(Y)+g(Y)Y]dY}

-5

g(Y)HKdY

= 0 (8.7)

These equations constitute a system of ordinary differential equations for axial velocity W and circulation K. Approximating