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Vortex breakdown as a fundamental element of vortex dynamics
Fluid Dynamics North-Holland
Research
3 (1988) 31-42
31
Vortex breakdown as a fundamental element of vortex dynamics J.J. KELLER, Brown
Bow-i
W. E3GLI and R. ALTHAUS
Research
Center, 5405 Baden.
Switzerland
Abstract. This paper presents an extension of an earlier analysis on vortex breakdown in tubes of constant cross section by Keller et al. (1985) to axisymmetric vortex flows in tubes with varying cross-sectional area. A broad investigation into the properties of steady axisymmetric vortex flows is given, insofar as they can be represented by perfect-fluid theory and simple extensions of it. It is argued that the basic physics of such flows, including vortex breakdown phenomena, can be explained with the help of the relatively simple theoretical concepts proposed. Numerical results are presented which show complete flow fields of loss-free transitions in diffusers.
1. Introduction Keller et al. (1985) have shown that in analogy to the theory of gravity currents in closed channels there exist two types of force-free transitions between axisymmetric flow states in tubes. Transitions of the first type lead directly to subcritical vortices and are associated with dissipation, whereas a transition of the second type incorporates a discrete loss-free transition leading to a supercritical hollow-core vortex which can be followed by a “vortex-hydraulic jump” finally leading to a subcritical vortex. For certain downstream boundary conditions both types of transitions are possible. In this case the history of the vortex flow determines which of the two types will appear. Although the underlying analysis, which is based on a variational principle, does not account for viscous effects or shear-layer instability, it does nevertheless explain the basic physics of vortex breakdown. In the subsequent series of photographs vortex breakdown structures (2a,b,c) are compared to the corresponding situations for gravity currents (la,b,c). The flow direction is always from left to right and the Reynolds number for the vortex-flow cases is about 5000. The first two photographs (figs. la, 2a) show loss-free transitions to supercritical flows. In the case of the gravity currents the bubble is air-filled. For this reason a flow with a free surface appears after the transition. The second pair of photographs (lb,2b) show a hydraulic jump moving upstream towards the loss-free transition. In the hydraulic case it is caused by a rising weir downstream, whereas in the case of the vortex flow an upstream moving orifice is responsible for the approaching vortex-hydraulic jump. Finally, the loss-free transition and the hydraulic jump combine to a single dissipative transition which leads directly to subcritical flow. At this stage the whole transition begins to respond to the imposed downstream boundary condition (see third pair of photographs). The analogy between the loss-free transitions becomes even closer if the vortex-breakdown bubble is also air-filled (compare figs. la and 4) or if the air in the case of the loss-free gravity current is replaced by a fluid with almost the same density as that of the running water (compare figs. 2a and 3). Since the density of the bubble fluid is somewhat larger than that of the running water, the gravity current bubble appears at the bottom of the channel. The close analogy between gravity currents and vortex-breakdown structures is of course the result of the 0169-5983/88/S4.00
0 1988, The Japan
Society of Fluid
Mechanics
J.J. Keller et ul. / Vortex hreukdown
32
Fig. 1. Transitions of water flows in a channel with a rectangular Force- and loss free transition to a flow with a free surface.
cross section
(height:
40 mm. breadth:
125 mm). (a)
analogy between gravity and centrifugal acceleration. Using the term VORTEX-BREAKDOWN we feel obliged to define its meaning in the present context: Presently “vortex breakdown” is defined to be a spontaneous transition from a supercritical to a subcritical vortex-flow state which can incorporate more than one stage. It may be interesting to note that
Fig. 1. (b) Transient:
first transition
and approaching
hydraulic
jump
marked
by an arrow
J.J. Keller et al. / Vortex breakdown
the
Fig. 1. (c) Combined
transition.
is used
a large
VORTEX-BREAKDOWN
term
dynamics
which is probably
investigation vortex-flow
represents
the main reason
an extension
states in tubes (Keller
for
variety
for the confusion
of our theory of force-
et al. 1985)
33
of phenomena
about
and loss-free
to the case of varying
in vortex
this term. The present transitions
cross-sectional
between area. The
idea is to provide a simple and elegant theoretical tool to discuss axisymmetric vortex flows - a theory which should be comparable in some sense to quasi-one-dimensional gasdynamics in the complex
field of compressible
structures
appear
particular
attention
subcritical structures.
The
through
theory
predicts
tubes of varying
must be paid to the critical-flow
how and where cross-sectional
cross sections
vortex-breakdown
area. In this context
which largely determine
the
velocity profiles and, as a consequence, the type and size of vortex-breakdown The jet- or wake-like character of the axial velocity profile in the core domain of a
Fig. 2. LIF-photographs Jump.
flows.
and propagate
showing
vortex
breakdown
in a 20”-diffuser.
(a) First transItIon
separated
from hydraulic
Fig. 2. (b) Transient:
first transitlon
and approaching
Fig. 2. (c) Combined
hydraulic
jump
marked
by an arrow.
transltion
subcritical vortex is always controlled by a downstream critical cross section. Furthermore, jet-like velocity profiles in supercritical flows lead to particularly large breakdown bubbles, whereas the opposite is true for wake-like velocity profiles. To include such aspects in the
Fig. 3. Supercritical
gravity
current
in a rectangular
channel
(Courtesy
of %I. Wright).
J.J. Keller et al. / Vortexbreakdown
Fig. 4. Air-filled
vortex-breakdown
bubble
35
in a water flow
subsequent discussion it is convenient but not necessary to restrict the consideration to a specific class of vortex-flows by defining a special set of generating functions. The simplest possibility is to consider the complete class of vortices which correspond to the generating functions of Rankine vortices. This class includes all Rankine vortices and all vortices which are obtained by contraction or expansion of a certain Rankine vortex which will be called reference flow state. In particular the class includes vortices with jet- or wake-like cores.
2. The Rankine vortex as a flow state of reference
Hence, as illustrated in fig. 5, we consider a vortex flow with a reference flow state which can be represented as a Rankine vortex. We assume that somewhere upstream there is a cylindrical section of the vortex tube where the flow becomes independent of the axial coordinate. This flow state”. Furthermore, we assume that after a part of the vortex flow is called “up-stream
flow of
hollow
state
reference
Ronkine vortex
uj2(r)
flow
state
intermediate flow
state
- core
vortex
u2(r)
downstream flow
state
Fig. 5. Loss-free transition between flow states in a tube with a change of cross-sectional area, for generating H and I defined on the basis of a reference flow state which can be represented as a Rankine vortex.
functions
36
J.J. Keller rr ul. / Vorttcx hrecrkdown
certain change of cross-sectional area of the tube there is again a cylindrical section downstream flow state is reached. First we define the reference flow state by u(r)
w(r)
= o;,
or
if
r 5 r.
ar,f/r
if
r,, < r < R,, ’
=
where the
(1)
where u and w denote the axial and azimuthal velocity components, I!_&and w are constants. The equation of motion can be written in the form (see Squire (1956))
(2) where
II/ is the stream
function
defined
by
and u is the radial velocity component. Due to the conservation of total head and circulation along stream surfaces there exist generating functions H and Z which depend on the values of the stream function 4 only. Based upon the definition of the reference flow state and the stream function we obtain the explicit expressions for H and Z as functions of +
(3)
(4) where k is defined by k = 2w/U,,. Following Benjamin (1962) we can deduce a criterion for subcritical or supercritical upstream flow (i.e. flow state 1: see fig. 5) with the help of Sturm’s fundamental comparison theorem:
(r,/R, )*
+
1 - (r,/R,)*
ul’q’-l +~i(r;,/r,)‘~l](kr,/2)‘;0 ( ro/rl )’ - 1 I
<
supercritical for critical
flow,
(5)
subcritical where ikrlJo(krl)
J, ( kr, ) and C’, is the axial velocity in the potential criterion for a critical Rankine vortex.
3. The limit of thin vortex cores: t-,/R,
flow domain.
This is an extension
of Benjamin’s
-a 1
In the interest of a transparent consideration the discussion is now restricted to the limit of thin vortex cores. We assume that a certain change of the cross-sectional area (a contraction or an expansion) of the vortex tube leads from the upstream flow state 1 to an intermediate
J.J. Keller et al. / Vorrex breakdown
37
pre-breakdown flow state 12 which satisfies exactly the condition for breakdown. Furthermore, we assume that vortex breakdown - or more precisely, a force- and loss-free transition to a supercritical hollow-core vortex - occurs in the cylindrical domain of the vortex tube between the flow states 12 and 2. Thus far the distinction between the flow states 1 and 12 has a formal meaning only. Their connections to the reference flow states are essentially equivalent. However, as soon as the complete flow field is considered the criteria for existence and uniqueness of the supercritical states 1 and 12 turn out to be different. Following Keller et al. (1985) it is straightforward to obtain the swirl-number criterion for vortex breakdown in the limit of thin cores. We obtain the relation f(kr,)*
= (lJ,,/Oo)2
= ( R,,/R2)4
= l/s,“,
where s2 = R,/R,.
The corresponding criterion for vortex breakdown after the first flow state but ahead of the area change is obtained by simply replacing U,, by U,, R, by R, and s2 by s,. If s2 > s, (i.e. the case of an expanding vortex tube) the breakdown criteria are strictly applicable to both flow states. For very small swirl numbers the flow remains supercritical throughout the vortex tube without undergoing vortex breakdown. As the swirl number increases vortex breakdown first appears after the second flow state when the swirl number reaches the value kr, = fi/sf. When the swirl number kr, is increased further the vortex breakdown structure stabilizes in the diffuser between the two flow states so long as fi/s,’ < kr,, < fi/sf. As soon as kr, exceeds the upper limit of this interval the breakdown structure moves further upstream beyond the first flow state. In this case the sequence of events is rather simple and obvious. However, if s2
(7)
J.J. Keller et (11. / Vortex hreukdown
domain of subcritica upstream flow
Fig. 6. Swirl ratio 8, defined normalized defined
and a parameter
velocity
by (7) versus A, on the axis
by (8) for downstream
conditions values
excess and
for
various
breakdown contraction
s.
A,,
which is a measure for the jet- or wake-like character of the upstream axial velocity profile can construct a criterion for vortex breakdown between locations 12 and 2 of the form
P, =f(A,;
s>, where
S = sJs,.
we
(9)
This criterion relates the swirl number to the profile character and the area change defined by the ratio of radii S appears as a geometric parameter. Fig. 6 shows the swirl number p, as a function of the profile parameter A, for various ratios of radii S. For large values A, (i.e. strongly jet-like velocity profiles) the criterion becomes independent of the area change. Furthermore, fig. 6 shows the critical limit for the upstream flow state. Points on the subcritical side of the critical limit do not correspond to real flows if the existence of a local maximum of cross-sectional area between the flow states 1 and 2 can be excluded. Finally, fig. 6 also shows that downstream of a strong contraction vortex breakdown can occur for upstream flows only with strongly jet-like axial velocity profiles.
4. The solid-body
vortex
We can now proceed to the general case of axisymmetric flows. Among the flows with cores of arbitrary thickness the subclass of flows whose reference state can be represented as a solid-body vortex deserves a special consideration. Keller et al. (1985) have shown that breakdown of a solid-body vortex corresponds to a degenerate limit of breakdown of a Rankine vortex. In the limit of the solid-body vortex nonlinearity in the governing equation disappears. The consequences are that _ the critical swirl number and the swirl number at which vortex breakdown occurs become identical, _ the breakdown bubble disappears and degenerates to a stagnation line _ the thickness or, in other words, the axial extent of the vortex-breakdown transition becomes infinitely large.
J.J. Keller et ul. / Vortex breakdown
upstream
solid-
flaw
body
state
downstream
transition
flow
state
vortex loss-free in the
Fig. 7. Loss-free
39
in a diffuser
downstream
transition diffuser
hollow-core
of a flow state which can be represented
vortex as a solid-body
vortex
This peculiar degeneracy appears as a result of the missing nonlinearity and the missing flow force in a tube of constant cross section. However, in the more general case of varying cross-sectional area the tube walls exert flow forces which cancel the degeneracy even in the absence of nonlinearity. It may be interesting to note that in the appendix of a paper by Benjamin and Barnard (1964) Fraenkel tried to support the views expressed in the main presentation, that force- and loss-free transitions to hollow-core vortices generally do nor exist, by considering the special case of a solid-body vortex in a tube of constant cross section. While Fraenkel’s proof appears to be correct, it only applies to the exceptionally degenerate case he considered. We consider the situation illustrated in fig. 7 and assume for simplification that the flow at the upstream end of the diffuser can be represented as a solid-body vortex and that a loss-free transition to a hollow-core vortex appears in the diffuser. The general x-independent solution of the governing equation for a flow with a solid-body vortex as a reference state is $/(r)=:u,r2+r[AJ,(kr)+BY,(kr)],
(10)
where k = 2w/lJ, and A and B are constants which have to be chosen suitably to satisfy the boundary conditions. For a large increase of cross-sectional area an elementary vortex-flow theory leads to a “diffuser paradox”. For large swirl numbers it appears to be impossible to find steady axisymmetric solutions. However, this apparent paradox can be resolved by including the possibility of a loss-free transition to a hollow-core vortex. As a first step towards a complete solution for the diffuser flow field we consider the flow state downstream of the diffuser. Following the elementary theory of axisymmetric vortex flows (see Batechelor (1967)) we obtain an expression for the stream function # = IJ> describing the downstream hollow-core vortex (see eq. (10)) where A=A
_
’ B=B,=
U,_ (R:-R:)Yt(kr,)+r,R,Y,(kR,) 2R,
J,(kR,)Y,(kra)
U, ___ 2R,
(R:-
-J,(kr.)Y,(kR,)
R:)J,(kra)
J,(kR,)Y,(kr,)
+raR,J,(kR,)
’
(11)
-J,(kr,)Y,(kR,).
With the exception of the bubble radius ra all parameters of the downstream flow state are now determined for any given swirl number. The underlying variational principle for general axisymmetric flows, discussed by Keller et al. (1985) leads to a natural boundary condition along the breakdown bubble surface which implies that all velocity components vanish at the bubble surface. In particular it implies that u~(T_,) = U, +A,/%.&,( kr,) + B2kYo( kra) = 0.
(12)
40
I-3.832
0 Fig. 8. Dimensionless
1 downstream
1
0
-2
-1
-2
-1
Fig, 9. Streamline maps of transitions represented as a solid-body vortex
2
bubble
{/a
0
I
0
1
3
2
2
for various
!z
versus swirl number
5
4
3
2
in a diffuser
radius
7
6
3
4
3
4
swirl numbers.
4
( for various
X
8
5
6
5
6
values S.
x
x
7
In each case the upstream
flow can br
41
J.J. Keller et al. / Vortex breakdown
Introducing the expressions for A, and B, in this condition yields a simple relation of the form 5 =f(.$; S) where { = kr;, is the dimensionless bubble radius, 5 = kR, the swirl number and S = RJR, the geometric parameter for the diffuser. For every given value S there exists an for the swirl number < in which the corresponding loss-free transition interval ~,i, 5 6 < t,,, to a hollow-core vortex stabilizes in the diffusers. The upper limit of the interval is determined by the condition for breakdown of a solid-body vortex which is equivalent to the condition for critical flow in this particular case < nldX= j, = 3.832,
(13)
where j, denotes the smallest positive zero of the Bessel function J,. To determine the lower we make use of the fact that in the absence of flow forces and nonlinearity the limit E,,, breakdown bubble degenerates to a stagnation line on the axis. In this limit the expression describing the downstream flow state reduces to s=
I - ~~~,(rz,i”)]-“‘,
(14)
] which is an implicite expression for the lower interval limit E,,,. Fig. 8 shows the dimensionless bubble radius t-JR, = l/q plotted versus the swirl number 5 for various ratios of radii S = R,/R,. The values tmin are reached at points where l/q vanishes. Having determined the downstream hollow-core flow states the complete flow fields are computed with the help of the numerical method which has been used by Keller et al. (1985) to compute transitions in tubes of constant cross section. The key idea is that the governing equation is solved with interchanged variables using a Jacobi relaxation method. Fig. 9 shows the complete flow fields in a sinusoidal diffuser (upstream and downstream ends are marked by arrows) for different swirl numbers 5. Similar results are presented for more general axisymmetric vortex flows.
5. General axisymmetric
vortex-flows
In this section we consider the general problem of loss-free transitions in axisymmetric flows. To compute the complete flow field between an upstream flow state 1 and a downstream flow state 2 we proceede as follows: For convenience we normalize the stream function, such that 1c,= 1 corresponds to the tube the generating wall. Assuming that the upstream flow state is given, we first determine functions H( 4) and I( $1) (see Batchelor (1967, Sect. 7.5)). Secondly we solve the x-indepenvariables. dent equation of motion (see eq. (2)) with interchanged
?e+H’(l))
--
I’(G)
2y
=O,
whereY=ir2,
(15)
Y;
at location
2 (i.e. for the downstream Y(l) = fR;,
y(0) = 0
y(l)=+R:,
ypcc
flow state) for the boundary
conditions (16)
and/or as$+O.
(17)
If the swirl number of the upstream flow state is sufficiently small (5 < EC, say) there exists a unique solution of (15) which satisfies the boundary conditions (16). In this case vortex breakdown does not occur between the flow states. If the swirl number E exceeds a certain value 5,,, but still remains smaller than &, it is possible that acceptable solutions of (15) exist
42
J.J. Keller et 01. / Vortex breakdown
for both sets of boundary conditions (16) and (17). The first set of boundary conditions corresponds to vortex flows without vortex breakdown between the two flow states whereas a loss-free transition to a hollow-core vortex appears between the flow states 1 and 2 if (17) applies. Finally, there is a third value 5 = E,,, which represents the upper swirl number limit for transitions to appear between the flow states 1 and 2. It should be pointed out that .& - E,,,, is nonnegative, whilst E,,,, - 6, can be positive or negative. If the variation of cross-sectional area between the two flow states corresponds to a diffuser with a sufficiently large ratio of radii R/R,, 5.,;,, - 5, is positive. However, E,,,;,, - 6, can be negative for very small values RJR,. Hence in the interval
E,,,
5
Es mid 5,,, , E,)
two solutions can be constructed. The history of the flow field and the flow conditions downstream of the second flow state decide which of the two solutions will appear. If E = 5, and the first set of boundary conditions applies the downstream flow state becomes just critical. No solution (diffuser paradox) of (15) can be found which satisfies (16) if 5 > EC. For E > t”,,,, it is still possible to find a solution of (15) which satisfies (17). However, in this case it is not possible to connect the two flow states with a flow field which satisfies the equation of motion (2). Hence, if E > max(<,, E,,,) the upstream flow field is not compatible with the geometry of the uortex tube between locutions I und 2. If the definition of the upstream flow field is directly based on the geometry of a swirl generator. for example. which is installed at location 1, we have to include the possibility of a hollow-core vortex emerging directly from the swirl generator. This situation is typical, for example, for the flow fields downstream of axial turbines. Having determined a downstream flow state the corresponding flow field between the flow states 1 and 2 is again computed by means of the numerical method proposed by Keller et al, (1985).
References Batch&r, G.K. (1967) An introduction to fluid dynamics. (Cambridge University Press, Cambridge) Benjamin, T.B. (1962) Theory of the vortex breakdown phenomenon, J. Flurd Mech. 14, 593. Benjamin, T.B. and Barnard B.J.S. (1964) A study of the motion of a cavity in a rotating liquid, J. Fluid Mech. IY, 193. Keller, J.J., Egli, W. and E&y, J. (1985) Force- and lossfree transitions between flow states. Z. anger. Math. Ph.vs. 36. 854. Squire, H.B. (1956) Rotating fluids, Surveys in Mechanics. ed. Batchelor and Davies (Cambridge Univ. Press, Cambridge).
Research
3 (1988) 31-42
31
Vortex breakdown as a fundamental element of vortex dynamics J.J. KELLER, Brown
Bow-i
W. E3GLI and R. ALTHAUS
Research
Center, 5405 Baden.
Switzerland
Abstract. This paper presents an extension of an earlier analysis on vortex breakdown in tubes of constant cross section by Keller et al. (1985) to axisymmetric vortex flows in tubes with varying cross-sectional area. A broad investigation into the properties of steady axisymmetric vortex flows is given, insofar as they can be represented by perfect-fluid theory and simple extensions of it. It is argued that the basic physics of such flows, including vortex breakdown phenomena, can be explained with the help of the relatively simple theoretical concepts proposed. Numerical results are presented which show complete flow fields of loss-free transitions in diffusers.
1. Introduction Keller et al. (1985) have shown that in analogy to the theory of gravity currents in closed channels there exist two types of force-free transitions between axisymmetric flow states in tubes. Transitions of the first type lead directly to subcritical vortices and are associated with dissipation, whereas a transition of the second type incorporates a discrete loss-free transition leading to a supercritical hollow-core vortex which can be followed by a “vortex-hydraulic jump” finally leading to a subcritical vortex. For certain downstream boundary conditions both types of transitions are possible. In this case the history of the vortex flow determines which of the two types will appear. Although the underlying analysis, which is based on a variational principle, does not account for viscous effects or shear-layer instability, it does nevertheless explain the basic physics of vortex breakdown. In the subsequent series of photographs vortex breakdown structures (2a,b,c) are compared to the corresponding situations for gravity currents (la,b,c). The flow direction is always from left to right and the Reynolds number for the vortex-flow cases is about 5000. The first two photographs (figs. la, 2a) show loss-free transitions to supercritical flows. In the case of the gravity currents the bubble is air-filled. For this reason a flow with a free surface appears after the transition. The second pair of photographs (lb,2b) show a hydraulic jump moving upstream towards the loss-free transition. In the hydraulic case it is caused by a rising weir downstream, whereas in the case of the vortex flow an upstream moving orifice is responsible for the approaching vortex-hydraulic jump. Finally, the loss-free transition and the hydraulic jump combine to a single dissipative transition which leads directly to subcritical flow. At this stage the whole transition begins to respond to the imposed downstream boundary condition (see third pair of photographs). The analogy between the loss-free transitions becomes even closer if the vortex-breakdown bubble is also air-filled (compare figs. la and 4) or if the air in the case of the loss-free gravity current is replaced by a fluid with almost the same density as that of the running water (compare figs. 2a and 3). Since the density of the bubble fluid is somewhat larger than that of the running water, the gravity current bubble appears at the bottom of the channel. The close analogy between gravity currents and vortex-breakdown structures is of course the result of the 0169-5983/88/S4.00
0 1988, The Japan
Society of Fluid
Mechanics
J.J. Keller et ul. / Vortex hreukdown
32
Fig. 1. Transitions of water flows in a channel with a rectangular Force- and loss free transition to a flow with a free surface.
cross section
(height:
40 mm. breadth:
125 mm). (a)
analogy between gravity and centrifugal acceleration. Using the term VORTEX-BREAKDOWN we feel obliged to define its meaning in the present context: Presently “vortex breakdown” is defined to be a spontaneous transition from a supercritical to a subcritical vortex-flow state which can incorporate more than one stage. It may be interesting to note that
Fig. 1. (b) Transient:
first transition
and approaching
hydraulic
jump
marked
by an arrow
J.J. Keller et al. / Vortex breakdown
the
Fig. 1. (c) Combined
transition.
is used
a large
VORTEX-BREAKDOWN
term
dynamics
which is probably
investigation vortex-flow
represents
the main reason
an extension
states in tubes (Keller
for
variety
for the confusion
of our theory of force-
et al. 1985)
33
of phenomena
about
and loss-free
to the case of varying
in vortex
this term. The present transitions
cross-sectional
between area. The
idea is to provide a simple and elegant theoretical tool to discuss axisymmetric vortex flows - a theory which should be comparable in some sense to quasi-one-dimensional gasdynamics in the complex
field of compressible
structures
appear
particular
attention
subcritical structures.
The
through
theory
predicts
tubes of varying
must be paid to the critical-flow
how and where cross-sectional
cross sections
vortex-breakdown
area. In this context
which largely determine
the
velocity profiles and, as a consequence, the type and size of vortex-breakdown The jet- or wake-like character of the axial velocity profile in the core domain of a
Fig. 2. LIF-photographs Jump.
flows.
and propagate
showing
vortex
breakdown
in a 20”-diffuser.
(a) First transItIon
separated
from hydraulic
Fig. 2. (b) Transient:
first transitlon
and approaching
Fig. 2. (c) Combined
hydraulic
jump
marked
by an arrow.
transltion
subcritical vortex is always controlled by a downstream critical cross section. Furthermore, jet-like velocity profiles in supercritical flows lead to particularly large breakdown bubbles, whereas the opposite is true for wake-like velocity profiles. To include such aspects in the
Fig. 3. Supercritical
gravity
current
in a rectangular
channel
(Courtesy
of %I. Wright).
J.J. Keller et al. / Vortexbreakdown
Fig. 4. Air-filled
vortex-breakdown
bubble
35
in a water flow
subsequent discussion it is convenient but not necessary to restrict the consideration to a specific class of vortex-flows by defining a special set of generating functions. The simplest possibility is to consider the complete class of vortices which correspond to the generating functions of Rankine vortices. This class includes all Rankine vortices and all vortices which are obtained by contraction or expansion of a certain Rankine vortex which will be called reference flow state. In particular the class includes vortices with jet- or wake-like cores.
2. The Rankine vortex as a flow state of reference
Hence, as illustrated in fig. 5, we consider a vortex flow with a reference flow state which can be represented as a Rankine vortex. We assume that somewhere upstream there is a cylindrical section of the vortex tube where the flow becomes independent of the axial coordinate. This flow state”. Furthermore, we assume that after a part of the vortex flow is called “up-stream
flow of
hollow
state
reference
Ronkine vortex
uj2(r)
flow
state
intermediate flow
state
- core
vortex
u2(r)
downstream flow
state
Fig. 5. Loss-free transition between flow states in a tube with a change of cross-sectional area, for generating H and I defined on the basis of a reference flow state which can be represented as a Rankine vortex.
functions
36
J.J. Keller rr ul. / Vorttcx hrecrkdown
certain change of cross-sectional area of the tube there is again a cylindrical section downstream flow state is reached. First we define the reference flow state by u(r)
w(r)
= o;,
or
if
r 5 r.
ar,f/r
if
r,, < r < R,, ’
=
where the
(1)
where u and w denote the axial and azimuthal velocity components, I!_&and w are constants. The equation of motion can be written in the form (see Squire (1956))
(2) where
II/ is the stream
function
defined
by
and u is the radial velocity component. Due to the conservation of total head and circulation along stream surfaces there exist generating functions H and Z which depend on the values of the stream function 4 only. Based upon the definition of the reference flow state and the stream function we obtain the explicit expressions for H and Z as functions of +
(3)
(4) where k is defined by k = 2w/U,,. Following Benjamin (1962) we can deduce a criterion for subcritical or supercritical upstream flow (i.e. flow state 1: see fig. 5) with the help of Sturm’s fundamental comparison theorem:
(r,/R, )*
+
1 - (r,/R,)*
ul’q’-l +~i(r;,/r,)‘~l](kr,/2)‘;0 ( ro/rl )’ - 1 I
<
supercritical for critical
flow,
(5)
subcritical where ikrlJo(krl)
J, ( kr, ) and C’, is the axial velocity in the potential criterion for a critical Rankine vortex.
3. The limit of thin vortex cores: t-,/R,
flow domain.
This is an extension
of Benjamin’s
-a 1
In the interest of a transparent consideration the discussion is now restricted to the limit of thin vortex cores. We assume that a certain change of the cross-sectional area (a contraction or an expansion) of the vortex tube leads from the upstream flow state 1 to an intermediate
J.J. Keller et al. / Vorrex breakdown
37
pre-breakdown flow state 12 which satisfies exactly the condition for breakdown. Furthermore, we assume that vortex breakdown - or more precisely, a force- and loss-free transition to a supercritical hollow-core vortex - occurs in the cylindrical domain of the vortex tube between the flow states 12 and 2. Thus far the distinction between the flow states 1 and 12 has a formal meaning only. Their connections to the reference flow states are essentially equivalent. However, as soon as the complete flow field is considered the criteria for existence and uniqueness of the supercritical states 1 and 12 turn out to be different. Following Keller et al. (1985) it is straightforward to obtain the swirl-number criterion for vortex breakdown in the limit of thin cores. We obtain the relation f(kr,)*
= (lJ,,/Oo)2
= ( R,,/R2)4
= l/s,“,
where s2 = R,/R,.
The corresponding criterion for vortex breakdown after the first flow state but ahead of the area change is obtained by simply replacing U,, by U,, R, by R, and s2 by s,. If s2 > s, (i.e. the case of an expanding vortex tube) the breakdown criteria are strictly applicable to both flow states. For very small swirl numbers the flow remains supercritical throughout the vortex tube without undergoing vortex breakdown. As the swirl number increases vortex breakdown first appears after the second flow state when the swirl number reaches the value kr, = fi/sf. When the swirl number kr, is increased further the vortex breakdown structure stabilizes in the diffuser between the two flow states so long as fi/s,’ < kr,, < fi/sf. As soon as kr, exceeds the upper limit of this interval the breakdown structure moves further upstream beyond the first flow state. In this case the sequence of events is rather simple and obvious. However, if s2
(7)
J.J. Keller et (11. / Vortex hreukdown
domain of subcritica upstream flow
Fig. 6. Swirl ratio 8, defined normalized defined
and a parameter
velocity
by (7) versus A, on the axis
by (8) for downstream
conditions values
excess and
for
various
breakdown contraction
s.
A,,
which is a measure for the jet- or wake-like character of the upstream axial velocity profile can construct a criterion for vortex breakdown between locations 12 and 2 of the form
P, =f(A,;
s>, where
S = sJs,.
we
(9)
This criterion relates the swirl number to the profile character and the area change defined by the ratio of radii S appears as a geometric parameter. Fig. 6 shows the swirl number p, as a function of the profile parameter A, for various ratios of radii S. For large values A, (i.e. strongly jet-like velocity profiles) the criterion becomes independent of the area change. Furthermore, fig. 6 shows the critical limit for the upstream flow state. Points on the subcritical side of the critical limit do not correspond to real flows if the existence of a local maximum of cross-sectional area between the flow states 1 and 2 can be excluded. Finally, fig. 6 also shows that downstream of a strong contraction vortex breakdown can occur for upstream flows only with strongly jet-like axial velocity profiles.
4. The solid-body
vortex
We can now proceed to the general case of axisymmetric flows. Among the flows with cores of arbitrary thickness the subclass of flows whose reference state can be represented as a solid-body vortex deserves a special consideration. Keller et al. (1985) have shown that breakdown of a solid-body vortex corresponds to a degenerate limit of breakdown of a Rankine vortex. In the limit of the solid-body vortex nonlinearity in the governing equation disappears. The consequences are that _ the critical swirl number and the swirl number at which vortex breakdown occurs become identical, _ the breakdown bubble disappears and degenerates to a stagnation line _ the thickness or, in other words, the axial extent of the vortex-breakdown transition becomes infinitely large.
J.J. Keller et ul. / Vortex breakdown
upstream
solid-
flaw
body
state
downstream
transition
flow
state
vortex loss-free in the
Fig. 7. Loss-free
39
in a diffuser
downstream
transition diffuser
hollow-core
of a flow state which can be represented
vortex as a solid-body
vortex
This peculiar degeneracy appears as a result of the missing nonlinearity and the missing flow force in a tube of constant cross section. However, in the more general case of varying cross-sectional area the tube walls exert flow forces which cancel the degeneracy even in the absence of nonlinearity. It may be interesting to note that in the appendix of a paper by Benjamin and Barnard (1964) Fraenkel tried to support the views expressed in the main presentation, that force- and loss-free transitions to hollow-core vortices generally do nor exist, by considering the special case of a solid-body vortex in a tube of constant cross section. While Fraenkel’s proof appears to be correct, it only applies to the exceptionally degenerate case he considered. We consider the situation illustrated in fig. 7 and assume for simplification that the flow at the upstream end of the diffuser can be represented as a solid-body vortex and that a loss-free transition to a hollow-core vortex appears in the diffuser. The general x-independent solution of the governing equation for a flow with a solid-body vortex as a reference state is $/(r)=:u,r2+r[AJ,(kr)+BY,(kr)],
(10)
where k = 2w/lJ, and A and B are constants which have to be chosen suitably to satisfy the boundary conditions. For a large increase of cross-sectional area an elementary vortex-flow theory leads to a “diffuser paradox”. For large swirl numbers it appears to be impossible to find steady axisymmetric solutions. However, this apparent paradox can be resolved by including the possibility of a loss-free transition to a hollow-core vortex. As a first step towards a complete solution for the diffuser flow field we consider the flow state downstream of the diffuser. Following the elementary theory of axisymmetric vortex flows (see Batechelor (1967)) we obtain an expression for the stream function # = IJ> describing the downstream hollow-core vortex (see eq. (10)) where A=A
_
’ B=B,=
U,_ (R:-R:)Yt(kr,)+r,R,Y,(kR,) 2R,
J,(kR,)Y,(kra)
U, ___ 2R,
(R:-
-J,(kr.)Y,(kR,)
R:)J,(kra)
J,(kR,)Y,(kr,)
+raR,J,(kR,)
’
(11)
-J,(kr,)Y,(kR,).
With the exception of the bubble radius ra all parameters of the downstream flow state are now determined for any given swirl number. The underlying variational principle for general axisymmetric flows, discussed by Keller et al. (1985) leads to a natural boundary condition along the breakdown bubble surface which implies that all velocity components vanish at the bubble surface. In particular it implies that u~(T_,) = U, +A,/%.&,( kr,) + B2kYo( kra) = 0.
(12)
40
I-3.832
0 Fig. 8. Dimensionless
1 downstream
1
0
-2
-1
-2
-1
Fig, 9. Streamline maps of transitions represented as a solid-body vortex
2
bubble
{/a
0
I
0
1
3
2
2
for various
!z
versus swirl number
5
4
3
2
in a diffuser
radius
7
6
3
4
3
4
swirl numbers.
4
( for various
X
8
5
6
5
6
values S.
x
x
7
In each case the upstream
flow can br
41
J.J. Keller et al. / Vortex breakdown
Introducing the expressions for A, and B, in this condition yields a simple relation of the form 5 =f(.$; S) where { = kr;, is the dimensionless bubble radius, 5 = kR, the swirl number and S = RJR, the geometric parameter for the diffuser. For every given value S there exists an for the swirl number < in which the corresponding loss-free transition interval ~,i, 5 6 < t,,, to a hollow-core vortex stabilizes in the diffusers. The upper limit of the interval is determined by the condition for breakdown of a solid-body vortex which is equivalent to the condition for critical flow in this particular case < nldX= j, = 3.832,
(13)
where j, denotes the smallest positive zero of the Bessel function J,. To determine the lower we make use of the fact that in the absence of flow forces and nonlinearity the limit E,,, breakdown bubble degenerates to a stagnation line on the axis. In this limit the expression describing the downstream flow state reduces to s=
I - ~~~,(rz,i”)]-“‘,
(14)
] which is an implicite expression for the lower interval limit E,,,. Fig. 8 shows the dimensionless bubble radius t-JR, = l/q plotted versus the swirl number 5 for various ratios of radii S = R,/R,. The values tmin are reached at points where l/q vanishes. Having determined the downstream hollow-core flow states the complete flow fields are computed with the help of the numerical method which has been used by Keller et al. (1985) to compute transitions in tubes of constant cross section. The key idea is that the governing equation is solved with interchanged variables using a Jacobi relaxation method. Fig. 9 shows the complete flow fields in a sinusoidal diffuser (upstream and downstream ends are marked by arrows) for different swirl numbers 5. Similar results are presented for more general axisymmetric vortex flows.
5. General axisymmetric
vortex-flows
In this section we consider the general problem of loss-free transitions in axisymmetric flows. To compute the complete flow field between an upstream flow state 1 and a downstream flow state 2 we proceede as follows: For convenience we normalize the stream function, such that 1c,= 1 corresponds to the tube the generating wall. Assuming that the upstream flow state is given, we first determine functions H( 4) and I( $1) (see Batchelor (1967, Sect. 7.5)). Secondly we solve the x-indepenvariables. dent equation of motion (see eq. (2)) with interchanged
?e+H’(l))
--
I’(G)
2y
=O,
whereY=ir2,
(15)
Y;
at location
2 (i.e. for the downstream Y(l) = fR;,
y(0) = 0
y(l)=+R:,
ypcc
flow state) for the boundary
conditions (16)
and/or as$+O.
(17)
If the swirl number of the upstream flow state is sufficiently small (5 < EC, say) there exists a unique solution of (15) which satisfies the boundary conditions (16). In this case vortex breakdown does not occur between the flow states. If the swirl number E exceeds a certain value 5,,, but still remains smaller than &, it is possible that acceptable solutions of (15) exist
42
J.J. Keller et 01. / Vortex breakdown
for both sets of boundary conditions (16) and (17). The first set of boundary conditions corresponds to vortex flows without vortex breakdown between the two flow states whereas a loss-free transition to a hollow-core vortex appears between the flow states 1 and 2 if (17) applies. Finally, there is a third value 5 = E,,, which represents the upper swirl number limit for transitions to appear between the flow states 1 and 2. It should be pointed out that .& - E,,,, is nonnegative, whilst E,,,, - 6, can be positive or negative. If the variation of cross-sectional area between the two flow states corresponds to a diffuser with a sufficiently large ratio of radii R/R,, 5.,;,, - 5, is positive. However, E,,,;,, - 6, can be negative for very small values RJR,. Hence in the interval
E,,,
5
Es mid 5,,, , E,)
two solutions can be constructed. The history of the flow field and the flow conditions downstream of the second flow state decide which of the two solutions will appear. If E = 5, and the first set of boundary conditions applies the downstream flow state becomes just critical. No solution (diffuser paradox) of (15) can be found which satisfies (16) if 5 > EC. For E > t”,,,, it is still possible to find a solution of (15) which satisfies (17). However, in this case it is not possible to connect the two flow states with a flow field which satisfies the equation of motion (2). Hence, if E > max(<,, E,,,) the upstream flow field is not compatible with the geometry of the uortex tube between locutions I und 2. If the definition of the upstream flow field is directly based on the geometry of a swirl generator. for example. which is installed at location 1, we have to include the possibility of a hollow-core vortex emerging directly from the swirl generator. This situation is typical, for example, for the flow fields downstream of axial turbines. Having determined a downstream flow state the corresponding flow field between the flow states 1 and 2 is again computed by means of the numerical method proposed by Keller et al, (1985).
References Batch&r, G.K. (1967) An introduction to fluid dynamics. (Cambridge University Press, Cambridge) Benjamin, T.B. (1962) Theory of the vortex breakdown phenomenon, J. Flurd Mech. 14, 593. Benjamin, T.B. and Barnard B.J.S. (1964) A study of the motion of a cavity in a rotating liquid, J. Fluid Mech. IY, 193. Keller, J.J., Egli, W. and E&y, J. (1985) Force- and lossfree transitions between flow states. Z. anger. Math. Ph.vs. 36. 854. Squire, H.B. (1956) Rotating fluids, Surveys in Mechanics. ed. Batchelor and Davies (Cambridge Univ. Press, Cambridge).