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A contribution to the problem of vortex breakdown
Computers & Fluids Vol. 13, No. 3, pp. 375-381, 1985 Printed in the U.S.A.
0045-7930/85 $3.00 + .00 © 1985 Pergamon Press Ltd.
A CONTRIBUTION TO THE PROBLEM OF VORTEX BREAKDOWN E. KRAUSE Aerodynamisches Institut der RWTH Aachen Aachen, Germany (Received 8 March 1984)
Abstract--Associated with the breakdown process is the formation of a stagnation point on the axis of the vortex. This requires the deceleration of the axial velocity component, which must be enforced by a positive axial pressure gradient. The analysis presented here shows, how the pressure gradient along the axis of the vortex is influenced by the radial and azimuthal velocity components. An explicit expression for Op/Ox (x, 0) can be obtained by integration of the momentum equation for the radial velocity component with respect to the radial and subsequent differentiation of the integral with respect to the axial direction. In an order of magnitude analysis it is then demonstrated that for large Reynolds numbers one component of the frictional force in the azimuthal direction cannot be neglected. In order to obtain an estimate for the pressure gradient rigid body rotation is assumed for the vortex core, and a distribution similar to that of a potential vortex w = kr-", for the outer portion. The estimate shows that a positive axial pressure gradient can exist only, if the radial velocity component is positive and if the exponent n is less than unity. It is also verified that a potential vortex cannot support an axial pressure gradient, that the pressure gradient in magnitude is directly proportional to the square of the maximum of the azimuthal velocity, referenced to the freestream velocity. 1. I N T R O D U C T I O N In [ 1] S. Leibovich c o m m e n t s on the effects o f the pressure gradients on vortex breakdown: "Vortex breakdown is p r o m o t e d by adverse pressure gradients (pressure increasing in the direction o f flow). Pressure gradients m a y be impressed u p o n the vortex core by an acceleration o f the outer flow (e.g. by shaping the walls o f a confining tube); from pressure differences along the vortex core caused by a sudden expansion if the tube abruptly ends; or, in the leading-edge vortex, from the pressure rise associated with the trailing edge o f a swept wing." M. G. Hall summarizes in [2] (as quoted in [1]) the role o f an applied pressure gradient: When an adverse pressure gradient is increased, less swirl is required to maintain breakdown, or, if the same level of swirl is maintained, the breakdown is moved upstream. It is this point which is investigated in this paper. The analysis was motivated by a numerical study, reported in [3]. In that study, it was assumed that the radial distribution o f the axial, radial and azimuthal velocity c o m p o n e n t s were k n o w n at some initial station. T h e equations o f m o t i o n were simplified in the spirit o f Prandtl's boundary-layer theory and integrated with an implicit finite-difference solution under the assumption o f axisymmetric flow. T h e results o f the integration showed that the axial velocity c o m p o n e n t a p p r o a c h e d zero with increasing distance from the initial station if certain specific initial distributions o f the velocity vector were prescribed. For very small values o f u(x, r = 0), (u being the axial velocity c o m p o n e n t ) the solution did not converge any more, yielding physically unrealistic results as in the vicinity o f boundary-layer separation. In addition, it was found, that if a positive pressure gradient was superimposed in the axial direction for r ~ oo, the solution diverged at a point closer to the initial station than for the zeropressure gradient flow. These results, although at least qualitatively k n o w n before the investigation [3] was carried out, for example, from Sarphaya's experiments [4], were also i m p l e m e n t e d by R. Staufenbiel in his experimental studies o f wing tip vortices [5]. By placing an a n n u l a r diffusor about the vortex he showed that he could locally break the vortex at a n y arbitrary position downstream from the wing. These observations as well as the c o m m e n t s m a d e by S. Leibovich and M. G. Hall and the results o f the numerical 375
376
E. KRAUSE
studies reported in [3], stipulated the question, as to whether or not this behaviour of the flow could not directly be confirmed by inspection of the governing equations, put into a suitable form. Despite of the rather large body of literature (see, for example, the literature survey in S. Leibovich's review on vortex breakdown [1]) such an investigation was--to the author's knowledge--not reported before. The development of an analysis which shows the dependence of the pressure gradient along the axis of the three velocity components is given in the following. It must be clear from the beginning, that the analysis does not intend to integrate the equations of motion in order to obtain a solution to the problem of vortex breakdown. Because of its highly non-linear character such a goal cannot be achieved without efforts; instead it is intended to extract general information from the governing equations in order to explain the overall behaviour of the flow and find out what simplifications are admissable upstream of the point of vortex breakdown. 2. ANALYSIS The flow upstream of the point of breakdown is assumed to be incompressible, steady and axisymmetric. The assumption of steady flow conditions is confirmed by experiments, see for example [2]; the assumption of axisymmetric flow implies that the azimuthal velocity component is essentially smaller than the axial component. As is also known from experiments the core of the vortex does, under such conditions, not change its diameter substantially, indicating that the radial velocity component is small and that the highest order derivatives in the axial direction can be neglected. It needs not be emphasized, that these assumptions become invalid downstream from the point of breakdown, as was, for example, also confirmed in [3], in addition to prior experimental evidence, reported in [1] and elsewhere. With the omission of the highest order terms in the axial direction, describing the corresponding components of the frictional force per unit volume, and the terms describing the variation in the azimuthal direction, the governing equations read: Continuity equation: 0 (ru)+ a 0-7 (rv) = o
(2.1)
Radial m o m e n t u m equation."
u Ox + v Or
r
q Or
Or
Orr (rv)
(2.2)
)]
(2.3)
Azimuthal m o m e n t u m equation:
U a x + V-gr + --r =
(rw)
Axial m o m e n t u m equation:
u ax + v Or
q ax
r Or \ -~rl.]"
(2.4)
All quantities have the usual meaning, with u, v, and w being the axial, radial, and azimuthal velocity components, respectively. Equations (2.1)-(2.4) are next rewritten in dimensionless form. In accord with the assumption stated above, it is justified to assume that the core radius R is much smaller than the distance L between the point of initiation and of breakdown of the vortex, i.e. R < L; consequently, the order of magnitude of v/u~ is ~ = R]L. The reference velocity for the azimuthal velocity component can be chosen arbitrarily. It can therefore be
377
The problem of vortex breakdown
expressed in terms of the undisturbed axial velocity components u~, say WR = aUo~, where the subscript R denotes the reference velocity, and ot some factor, either smaller or larger than unity. With the pressure normalized with pu~ 2 the nondimensionalized equations read 0 0 -~x (ru) + "~r (rv) = 0
( ov
(2.5)
)]
0,2 02w2
u-~x +v-&r}'
~--
Or
Re Or r ~ ( r v )
(2.6)
Ow Ow vw 1 I 0 (_lrO )] U -~x + v ~r + r - Ree 2 Or (rw)
u -~x + v Or
Ox
(2.7)
------5 L\ Yr Ret / Jr ~rr r
"
(2.8)
In the system of equations eqns (2.5)-(2.8), all variables are now dimensionless quantities, with v/u~ stretched by R / L . Equation (2.6) is now integrated with respect to r; there results p(x, O) = p(x, ~ ) +
u -~x + v -~rJt
-r J dr - -R-e J o
& r & ( r v ) dr.
(2.9)
By making use of the continuity equation, the last equation can be casted into the following form: a2w2-] p(x, o) - p(x, ~ ) =
u2 ~x
- r
~2 _ _ _r -Jdr + ~
l
?.
o.
~ x (x, ~ ) - ~ (x, o)
).
(2.10)
From eqn (2.10) it is clear that the vortex can exist only as long as the right-hand side remains negative. Vice versa, if the pressure difference and the gradients of the axial velocity component tend to zero, the azimuthal velocity component must vanish. If the flow along the axis is to approach a stagnation point, the pressure along the axis must increase, i.e. Op/ax(x, 0) > 0. An expression for the pressure gradient can be obtained by differentiating eqn (2.10) with respect to the axial coordinate. With the definitions ( A p ) r = p ( X , 0 ) --
p(X, o0)
(2.11)
(AU)r = U(X, O) -- U(X, o0)
there results OOx( Ap)r = J
fore2u ~xouo~x (u) + u2 ~0 It uv )\
2v Ov'] oxJ dr
- r
_ a2 L ~ 2w ow dr -7" O----x
1 O2 R e OX2 (Aid)r"
(2.12)
The axial gradient of the azimuthal velocity component can be replaced by eqn (2.7): Ow Ox
v 0 -I ur Or (rw) + -u- -Re,
0
(2.13)
378
E. KRAUSE
With eqn (2.13), eqn (2.12) becomes
-~xO( Ap) r = ~z
2U ~x -~x U + u2 0x 22a2
°~
-- --r Ox dr
1 02 Re Ox 2 (Au)r
r O ( l O (rw))]dr + 2az fo~ -v- -w- - O (rw)dr. r 2 Or
U
(2.14)
If the order of magnitude assumptions stated earlier, namely o~ < 1,
e2 < 1,
Re>> 1,
Re~ 2 = O(1)
(2.15)
and introduced in eqn (2.14), there is finally obtained
O(mp)r- Re,2°d2~O°°IwlO(lO ~WO 2 u r Or r O-r(rW))1dr + 2a 2 fOr~( -~-~r(rW) )dr.
~x
(2.16)
Equation (2.16) shows that to zeroth order the pressure gradient along the axis is influenced by both, convectional and viscous terms, represented by the two integrals on the right hand side of eqn (2.16). In fact, if the radial velocity component is assumed to be a slowly varying function of the axial distance x, and if v is close to zero initially, the second integral can completely be neglected. Then the increase of the pressure in the axial direction is solely built up by viscous forces (in the frame of the order of approximation considered here). Note that the factor 1/(Re~ 2) could have been absorbed by either setting it equal to unity or by coordinate stretching. It was left in the equations in order to indicate the contribution of the viscous forces. The influence of the first-order terms is contained in eqn (2.14). The order of magnitude requirements, given by eqn (2.15) are identical with boundary layer assumptions. If the term multiplied by ~2 is maintained in the approximation, also terms of order 1/Re must be carried along. The first order terms cause an increase of the pressure along the axis, if the first integral in eqn (2.14) is positive, and the second term is negative. The assumptions given by eqn (2.15) become invalid in the immediate vicinity of the stagnation point, i.e. the point where the breakdown is initiated. Hall's observation [1], quoted in the introduction, is clearly confirmed through eqn (2.16): The pressure gradient on the axis can be enforced either by a certain distribution of the azimuthal velocity component or by a pressure gradient imposed on the outer flow i.e.: ~
~ x ( X , oo) - 2 ~ 2
I, - h
•
(2.17)
The expressions I1 and /2 designate the two integrals given in eqn (2.16). If the pressure gradient on the axis is required to be positive, the right hand side of eqn (2.17) must be greater than zero. This can be enforced either by certain distributions of the azimuthal velocity components which are such that the bracketed term in eqn (2.17) is negative, while Op/Ox(x, 0) is zero, or for a non-vanishing positive pressure gradient the bracketed term can be smaller in magnitude. In other words, Hall's statement can be casted into the following form. If Ap(0) is the pressure difference along the axis necessary to cause breakdown, then Ap(0) = Ap(oo) - 2 a 2 [ ~ e ~2 ~ I I ~ d x - f
Izdx] = c o n s t a n t
(2.18)
In eqn (2.18) Ap(oo) is the pressure difference imposed on the external flow in the axial direction; the integration has to be extended over the length L, the distance between the point of initiation and of breakdown of the vortex. In order to investigate the conditions to be satisfied for breakdown, an estimate of the integrals is given in the next section.
The problem of vortex breakdown
379
3. ESTIMATE OF THE AXIAL PRESSURE GRADIENT Since it is impossible to extract further information from the integrals 11 and 12, additional assumptions must be introduced. The simplest assumption which can be made for the azimuthal velocity component near the axis is that the flow behaves like a rigid body, i.e., in dimensionless variables w = r, at least for r < r~, where rl is the dimensionless radius up to which rigid body rotation prevailes. The first integral Ii yields zero for r < r, while the second, 12, is
fo (: wo )
-~ Or (rw) dr = 4or2
In order to obtain an estimate of the integrals vortex, a distribution of the form
11
f0 --U
dr.
(3.1)
and /2 for the outer portion of the
w = kr -2n, n > 0
(3.2)
is assumed for the azimuthal component. The parts of the integrands containing w can then be evaluated by means of eqn (3.2). There is obtained
wO('rO ) r Or
O-r (rw)
= (n 2 - 1)kEr-(2n+3)
wO r E Or (rw) = - ( n - l)k2r -ztn+l),
(3.3)
so that the pressure gradient along the axis can be written as O---(AP)~ = 4°lZ fo~' ( V ) d r - 2(n
1)c~2k2 fr~
(V)r-2("+l)dr
- 2(n2-1)
a2k 2 fr ,~ u1 r_(2n+3)dr" Re¢---5
(3.4)
According to (3.4) the magnitude of the radial velocity component v and the exponent n, the radial decay of the azimuthal velocity, determine the sign of the pressure gradient. Note that the ratio v/u could be expressed by the continuity equation and the axial momentum equation v_ u
1 frOPr'dr r do Ox u 2
,
1 f0r 1 d ( Ou\ , - Re,2----r -~ Or--5, r' ~r,)dr.
(3.5)
Then, the pressure gradient appears also on the right-hand side of (3.4), and the sign is basically determined by the second integral in (3.5), which is more complicated than (3.4). No use is therefore made of (3.5). Although eqn (3.2) is restricted to a certain class of azimuthal velocity profiles, it could be extended to fit arbitrary profiles by choosing n variable and extending the last two integrals in (3.4) only over finite radial strips. This would yield a sum, in which in the exponent n would be constant only for every single strip. In the limit the sum could be replaced by an integration over the radial coordinate. In order to keep the analysis as simple as possible, this step is not introduced here. All important conclusions can be obtained from eqn (3.4). The influence of the inner part of the vortex, for which rigid body rotation is assumed is completely determined by the radial velocity component: If v is positive, then the first integral in (3.4) is always positive, while the opposite is true for a negative radial velocity component. The second expression in (3.4) depends on both, the exponent n of the radial decay for the azimuthal velocity component, and on the radial velocity. The following cases can be distinguished:
380
E. KRAUSE
Case l: The radial velocity component is positive and n is equal to one, so that the radial distribution of the azimuthal velocity component at every station x corresponds to that of a potential vortex. Of the three expressions on the right-hand side of (3.4) only the first is different from zero. The vanishing of the two latter expressions implies that a potential vortex cannot support an axial pressure gradient. For a positive normal velocity component the pressure gradient is also positive and vortex breakdown must occur. For a negative radial velocity component the axial flow is being accelerated. The magnitude of the axial velocity component is essential for the pressure gradient: If two vortices are compared with different initial axial velocity profiles, the one with the larger axial velocity component will encounter breakdown further downstream than the vortex in which the axial velocity is smaller.
Case 2: The exponent n is smaller than one. The azimuthal velocity component decays less than that of the potential vortex. If, as before, the integrals containing the radial velocity component are positive, all three expressions in (3.4) are positive so that the pressure gradient is larger than under case 1. Vortex breakdown must occur closer to the point of initiation of the vortex. If the first two integrals are negative, breakdown is delayed. In fact it may not occur at all.
Case 3: The exponent n is larger than one. The azimuthal velocity component has a steeper radial decay than that of the potential vortex. The last two expressions (3.4) are now negative if the second integral (containing v) is positive. Vortex breakdown is delayed, and does not occur if also the first integral is positive, i.e.
(n-1)k2 f~]lr-at"+l)(v + ~Re~ ] d r <] 2
for' (v)
(3.6)
If the first two integrals are negative, vortex breakdown is possible only if (rt--1)k 2
r-2t"+l)dr > 2
if0 (v) I dr
+ (n 2 -
l)~-~Se~ 2 , u
"
Also note, that the factor a 2, the ratio of the maximum azimuthal velocity and the freestream velocity can be included in the nondimensionalization of the pressure, so that p is referenced to qa2u~ 2, or ot2 can be absorbed by defining a new coordinate x = a2x. Then from the above relations, in particular (2.18), it is clear that the length L between the point of initiation and of breakdown of the vortex is inversely proportional to the square of the maximum azimuthal velocity and directly proportional to the square of the freestream velocity u~. This means that vortex breakdown can be delayed by increasing the former and decreasing the latter. 4. CONCLUDING REMARKS By casting the governing equations into a suitable form, it could be shown, how the pressure gradient along the axis of a vortex can be expressed by integral relations, containing the radial, axial, and azimuthal velocity components, and radial derivatives of the azimuthal velocity components. With simple assumptions for the radial distribution of the azimuthal velocity component it was demonstrated that not only inertia forces but also viscous forces contribute to the initiation of the breakdown process, which requires Op/Ox(x,O) to be positive. If the radius of the core is increasing in the direction of the axial flow, i.e. the radial velocity is always positive, breakdown must eventually occur. The analysis also shows, that the outer portion of the vortex cannot support a pressure
The problem of vortex breakdown
381
gradient along the axis, if the radial distribution of the azimuthal velocity c o m p o n e n t is equal to that of a potential vortex; that a positive pressure gradient Op/Ox(x, 0) results, if the radial decay of the azimuthal velocity components is less steep than that of the potential vortex for positive radial velocity components, and that an increase of the initial axial velocity c o m p o n e n t delays breakdown as does a decrease of the m a x i m u m azimuthal velocity component. The results given in this analysis a valid for large Reynolds numbers with small core radii. Extension to large core radii should be possible.
REFERENCES I. S. Leibovich, The Structure of Vortex Breakdown. Ann. Rev. Fluid Mech. 10, 221-246 (1978). 2. M. G. Hall, Vortex Breakdown. Ann. Rev. FluidMech. 4, 195-218 (1972). 3. X. Shi and E. Krause, "Numerische Untersuchung des Aufplatzens eines Wirbels." Paper presented at the GAMM Tagung 1983, Band 64, T230-232. 4. T. Sarphaya: On Stationary and Traveling Vortex Breakdown. J. Fluid Mech. 45, 545-559 {1971). 5. R. Staufenbiel: Private Communication (1982).
0045-7930/85 $3.00 + .00 © 1985 Pergamon Press Ltd.
A CONTRIBUTION TO THE PROBLEM OF VORTEX BREAKDOWN E. KRAUSE Aerodynamisches Institut der RWTH Aachen Aachen, Germany (Received 8 March 1984)
Abstract--Associated with the breakdown process is the formation of a stagnation point on the axis of the vortex. This requires the deceleration of the axial velocity component, which must be enforced by a positive axial pressure gradient. The analysis presented here shows, how the pressure gradient along the axis of the vortex is influenced by the radial and azimuthal velocity components. An explicit expression for Op/Ox (x, 0) can be obtained by integration of the momentum equation for the radial velocity component with respect to the radial and subsequent differentiation of the integral with respect to the axial direction. In an order of magnitude analysis it is then demonstrated that for large Reynolds numbers one component of the frictional force in the azimuthal direction cannot be neglected. In order to obtain an estimate for the pressure gradient rigid body rotation is assumed for the vortex core, and a distribution similar to that of a potential vortex w = kr-", for the outer portion. The estimate shows that a positive axial pressure gradient can exist only, if the radial velocity component is positive and if the exponent n is less than unity. It is also verified that a potential vortex cannot support an axial pressure gradient, that the pressure gradient in magnitude is directly proportional to the square of the maximum of the azimuthal velocity, referenced to the freestream velocity. 1. I N T R O D U C T I O N In [ 1] S. Leibovich c o m m e n t s on the effects o f the pressure gradients on vortex breakdown: "Vortex breakdown is p r o m o t e d by adverse pressure gradients (pressure increasing in the direction o f flow). Pressure gradients m a y be impressed u p o n the vortex core by an acceleration o f the outer flow (e.g. by shaping the walls o f a confining tube); from pressure differences along the vortex core caused by a sudden expansion if the tube abruptly ends; or, in the leading-edge vortex, from the pressure rise associated with the trailing edge o f a swept wing." M. G. Hall summarizes in [2] (as quoted in [1]) the role o f an applied pressure gradient: When an adverse pressure gradient is increased, less swirl is required to maintain breakdown, or, if the same level of swirl is maintained, the breakdown is moved upstream. It is this point which is investigated in this paper. The analysis was motivated by a numerical study, reported in [3]. In that study, it was assumed that the radial distribution o f the axial, radial and azimuthal velocity c o m p o n e n t s were k n o w n at some initial station. T h e equations o f m o t i o n were simplified in the spirit o f Prandtl's boundary-layer theory and integrated with an implicit finite-difference solution under the assumption o f axisymmetric flow. T h e results o f the integration showed that the axial velocity c o m p o n e n t a p p r o a c h e d zero with increasing distance from the initial station if certain specific initial distributions o f the velocity vector were prescribed. For very small values o f u(x, r = 0), (u being the axial velocity c o m p o n e n t ) the solution did not converge any more, yielding physically unrealistic results as in the vicinity o f boundary-layer separation. In addition, it was found, that if a positive pressure gradient was superimposed in the axial direction for r ~ oo, the solution diverged at a point closer to the initial station than for the zeropressure gradient flow. These results, although at least qualitatively k n o w n before the investigation [3] was carried out, for example, from Sarphaya's experiments [4], were also i m p l e m e n t e d by R. Staufenbiel in his experimental studies o f wing tip vortices [5]. By placing an a n n u l a r diffusor about the vortex he showed that he could locally break the vortex at a n y arbitrary position downstream from the wing. These observations as well as the c o m m e n t s m a d e by S. Leibovich and M. G. Hall and the results o f the numerical 375
376
E. KRAUSE
studies reported in [3], stipulated the question, as to whether or not this behaviour of the flow could not directly be confirmed by inspection of the governing equations, put into a suitable form. Despite of the rather large body of literature (see, for example, the literature survey in S. Leibovich's review on vortex breakdown [1]) such an investigation was--to the author's knowledge--not reported before. The development of an analysis which shows the dependence of the pressure gradient along the axis of the three velocity components is given in the following. It must be clear from the beginning, that the analysis does not intend to integrate the equations of motion in order to obtain a solution to the problem of vortex breakdown. Because of its highly non-linear character such a goal cannot be achieved without efforts; instead it is intended to extract general information from the governing equations in order to explain the overall behaviour of the flow and find out what simplifications are admissable upstream of the point of vortex breakdown. 2. ANALYSIS The flow upstream of the point of breakdown is assumed to be incompressible, steady and axisymmetric. The assumption of steady flow conditions is confirmed by experiments, see for example [2]; the assumption of axisymmetric flow implies that the azimuthal velocity component is essentially smaller than the axial component. As is also known from experiments the core of the vortex does, under such conditions, not change its diameter substantially, indicating that the radial velocity component is small and that the highest order derivatives in the axial direction can be neglected. It needs not be emphasized, that these assumptions become invalid downstream from the point of breakdown, as was, for example, also confirmed in [3], in addition to prior experimental evidence, reported in [1] and elsewhere. With the omission of the highest order terms in the axial direction, describing the corresponding components of the frictional force per unit volume, and the terms describing the variation in the azimuthal direction, the governing equations read: Continuity equation: 0 (ru)+ a 0-7 (rv) = o
(2.1)
Radial m o m e n t u m equation."
u Ox + v Or
r
q Or
Or
Orr (rv)
(2.2)
)]
(2.3)
Azimuthal m o m e n t u m equation:
U a x + V-gr + --r =
(rw)
Axial m o m e n t u m equation:
u ax + v Or
q ax
r Or \ -~rl.]"
(2.4)
All quantities have the usual meaning, with u, v, and w being the axial, radial, and azimuthal velocity components, respectively. Equations (2.1)-(2.4) are next rewritten in dimensionless form. In accord with the assumption stated above, it is justified to assume that the core radius R is much smaller than the distance L between the point of initiation and of breakdown of the vortex, i.e. R < L; consequently, the order of magnitude of v/u~ is ~ = R]L. The reference velocity for the azimuthal velocity component can be chosen arbitrarily. It can therefore be
377
The problem of vortex breakdown
expressed in terms of the undisturbed axial velocity components u~, say WR = aUo~, where the subscript R denotes the reference velocity, and ot some factor, either smaller or larger than unity. With the pressure normalized with pu~ 2 the nondimensionalized equations read 0 0 -~x (ru) + "~r (rv) = 0
( ov
(2.5)
)]
0,2 02w2
u-~x +v-&r}'
~--
Or
Re Or r ~ ( r v )
(2.6)
Ow Ow vw 1 I 0 (_lrO )] U -~x + v ~r + r - Ree 2 Or (rw)
u -~x + v Or
Ox
(2.7)
------5 L\ Yr Ret / Jr ~rr r
"
(2.8)
In the system of equations eqns (2.5)-(2.8), all variables are now dimensionless quantities, with v/u~ stretched by R / L . Equation (2.6) is now integrated with respect to r; there results p(x, O) = p(x, ~ ) +
u -~x + v -~rJt
-r J dr - -R-e J o
& r & ( r v ) dr.
(2.9)
By making use of the continuity equation, the last equation can be casted into the following form: a2w2-] p(x, o) - p(x, ~ ) =
u2 ~x
- r
~2 _ _ _r -Jdr + ~
l
?.
o.
~ x (x, ~ ) - ~ (x, o)
).
(2.10)
From eqn (2.10) it is clear that the vortex can exist only as long as the right-hand side remains negative. Vice versa, if the pressure difference and the gradients of the axial velocity component tend to zero, the azimuthal velocity component must vanish. If the flow along the axis is to approach a stagnation point, the pressure along the axis must increase, i.e. Op/ax(x, 0) > 0. An expression for the pressure gradient can be obtained by differentiating eqn (2.10) with respect to the axial coordinate. With the definitions ( A p ) r = p ( X , 0 ) --
p(X, o0)
(2.11)
(AU)r = U(X, O) -- U(X, o0)
there results OOx( Ap)r = J
fore2u ~xouo~x (u) + u2 ~0 It uv )\
2v Ov'] oxJ dr
- r
_ a2 L ~ 2w ow dr -7" O----x
1 O2 R e OX2 (Aid)r"
(2.12)
The axial gradient of the azimuthal velocity component can be replaced by eqn (2.7): Ow Ox
v 0 -I ur Or (rw) + -u- -Re,
0
(2.13)
378
E. KRAUSE
With eqn (2.13), eqn (2.12) becomes
-~xO( Ap) r = ~z
2U ~x -~x U + u2 0x 22a2
°~
-- --r Ox dr
1 02 Re Ox 2 (Au)r
r O ( l O (rw))]dr + 2az fo~ -v- -w- - O (rw)dr. r 2 Or
U
(2.14)
If the order of magnitude assumptions stated earlier, namely o~ < 1,
e2 < 1,
Re>> 1,
Re~ 2 = O(1)
(2.15)
and introduced in eqn (2.14), there is finally obtained
O(mp)r- Re,2°d2~O°°IwlO(lO ~WO 2 u r Or r O-r(rW))1dr + 2a 2 fOr~( -~-~r(rW) )dr.
~x
(2.16)
Equation (2.16) shows that to zeroth order the pressure gradient along the axis is influenced by both, convectional and viscous terms, represented by the two integrals on the right hand side of eqn (2.16). In fact, if the radial velocity component is assumed to be a slowly varying function of the axial distance x, and if v is close to zero initially, the second integral can completely be neglected. Then the increase of the pressure in the axial direction is solely built up by viscous forces (in the frame of the order of approximation considered here). Note that the factor 1/(Re~ 2) could have been absorbed by either setting it equal to unity or by coordinate stretching. It was left in the equations in order to indicate the contribution of the viscous forces. The influence of the first-order terms is contained in eqn (2.14). The order of magnitude requirements, given by eqn (2.15) are identical with boundary layer assumptions. If the term multiplied by ~2 is maintained in the approximation, also terms of order 1/Re must be carried along. The first order terms cause an increase of the pressure along the axis, if the first integral in eqn (2.14) is positive, and the second term is negative. The assumptions given by eqn (2.15) become invalid in the immediate vicinity of the stagnation point, i.e. the point where the breakdown is initiated. Hall's observation [1], quoted in the introduction, is clearly confirmed through eqn (2.16): The pressure gradient on the axis can be enforced either by a certain distribution of the azimuthal velocity component or by a pressure gradient imposed on the outer flow i.e.: ~
~ x ( X , oo) - 2 ~ 2
I, - h
•
(2.17)
The expressions I1 and /2 designate the two integrals given in eqn (2.16). If the pressure gradient on the axis is required to be positive, the right hand side of eqn (2.17) must be greater than zero. This can be enforced either by certain distributions of the azimuthal velocity components which are such that the bracketed term in eqn (2.17) is negative, while Op/Ox(x, 0) is zero, or for a non-vanishing positive pressure gradient the bracketed term can be smaller in magnitude. In other words, Hall's statement can be casted into the following form. If Ap(0) is the pressure difference along the axis necessary to cause breakdown, then Ap(0) = Ap(oo) - 2 a 2 [ ~ e ~2 ~ I I ~ d x - f
Izdx] = c o n s t a n t
(2.18)
In eqn (2.18) Ap(oo) is the pressure difference imposed on the external flow in the axial direction; the integration has to be extended over the length L, the distance between the point of initiation and of breakdown of the vortex. In order to investigate the conditions to be satisfied for breakdown, an estimate of the integrals is given in the next section.
The problem of vortex breakdown
379
3. ESTIMATE OF THE AXIAL PRESSURE GRADIENT Since it is impossible to extract further information from the integrals 11 and 12, additional assumptions must be introduced. The simplest assumption which can be made for the azimuthal velocity component near the axis is that the flow behaves like a rigid body, i.e., in dimensionless variables w = r, at least for r < r~, where rl is the dimensionless radius up to which rigid body rotation prevailes. The first integral Ii yields zero for r < r, while the second, 12, is
fo (: wo )
-~ Or (rw) dr = 4or2
In order to obtain an estimate of the integrals vortex, a distribution of the form
11
f0 --U
dr.
(3.1)
and /2 for the outer portion of the
w = kr -2n, n > 0
(3.2)
is assumed for the azimuthal component. The parts of the integrands containing w can then be evaluated by means of eqn (3.2). There is obtained
wO('rO ) r Or
O-r (rw)
= (n 2 - 1)kEr-(2n+3)
wO r E Or (rw) = - ( n - l)k2r -ztn+l),
(3.3)
so that the pressure gradient along the axis can be written as O---(AP)~ = 4°lZ fo~' ( V ) d r - 2(n
1)c~2k2 fr~
(V)r-2("+l)dr
- 2(n2-1)
a2k 2 fr ,~ u1 r_(2n+3)dr" Re¢---5
(3.4)
According to (3.4) the magnitude of the radial velocity component v and the exponent n, the radial decay of the azimuthal velocity, determine the sign of the pressure gradient. Note that the ratio v/u could be expressed by the continuity equation and the axial momentum equation v_ u
1 frOPr'dr r do Ox u 2
,
1 f0r 1 d ( Ou\ , - Re,2----r -~ Or--5, r' ~r,)dr.
(3.5)
Then, the pressure gradient appears also on the right-hand side of (3.4), and the sign is basically determined by the second integral in (3.5), which is more complicated than (3.4). No use is therefore made of (3.5). Although eqn (3.2) is restricted to a certain class of azimuthal velocity profiles, it could be extended to fit arbitrary profiles by choosing n variable and extending the last two integrals in (3.4) only over finite radial strips. This would yield a sum, in which in the exponent n would be constant only for every single strip. In the limit the sum could be replaced by an integration over the radial coordinate. In order to keep the analysis as simple as possible, this step is not introduced here. All important conclusions can be obtained from eqn (3.4). The influence of the inner part of the vortex, for which rigid body rotation is assumed is completely determined by the radial velocity component: If v is positive, then the first integral in (3.4) is always positive, while the opposite is true for a negative radial velocity component. The second expression in (3.4) depends on both, the exponent n of the radial decay for the azimuthal velocity component, and on the radial velocity. The following cases can be distinguished:
380
E. KRAUSE
Case l: The radial velocity component is positive and n is equal to one, so that the radial distribution of the azimuthal velocity component at every station x corresponds to that of a potential vortex. Of the three expressions on the right-hand side of (3.4) only the first is different from zero. The vanishing of the two latter expressions implies that a potential vortex cannot support an axial pressure gradient. For a positive normal velocity component the pressure gradient is also positive and vortex breakdown must occur. For a negative radial velocity component the axial flow is being accelerated. The magnitude of the axial velocity component is essential for the pressure gradient: If two vortices are compared with different initial axial velocity profiles, the one with the larger axial velocity component will encounter breakdown further downstream than the vortex in which the axial velocity is smaller.
Case 2: The exponent n is smaller than one. The azimuthal velocity component decays less than that of the potential vortex. If, as before, the integrals containing the radial velocity component are positive, all three expressions in (3.4) are positive so that the pressure gradient is larger than under case 1. Vortex breakdown must occur closer to the point of initiation of the vortex. If the first two integrals are negative, breakdown is delayed. In fact it may not occur at all.
Case 3: The exponent n is larger than one. The azimuthal velocity component has a steeper radial decay than that of the potential vortex. The last two expressions (3.4) are now negative if the second integral (containing v) is positive. Vortex breakdown is delayed, and does not occur if also the first integral is positive, i.e.
(n-1)k2 f~]lr-at"+l)(v + ~Re~ ] d r <] 2
for' (v)
(3.6)
If the first two integrals are negative, vortex breakdown is possible only if (rt--1)k 2
r-2t"+l)dr > 2
if0 (v) I dr
+ (n 2 -
l)~-~Se~ 2 , u
"
Also note, that the factor a 2, the ratio of the maximum azimuthal velocity and the freestream velocity can be included in the nondimensionalization of the pressure, so that p is referenced to qa2u~ 2, or ot2 can be absorbed by defining a new coordinate x = a2x. Then from the above relations, in particular (2.18), it is clear that the length L between the point of initiation and of breakdown of the vortex is inversely proportional to the square of the maximum azimuthal velocity and directly proportional to the square of the freestream velocity u~. This means that vortex breakdown can be delayed by increasing the former and decreasing the latter. 4. CONCLUDING REMARKS By casting the governing equations into a suitable form, it could be shown, how the pressure gradient along the axis of a vortex can be expressed by integral relations, containing the radial, axial, and azimuthal velocity components, and radial derivatives of the azimuthal velocity components. With simple assumptions for the radial distribution of the azimuthal velocity component it was demonstrated that not only inertia forces but also viscous forces contribute to the initiation of the breakdown process, which requires Op/Ox(x,O) to be positive. If the radius of the core is increasing in the direction of the axial flow, i.e. the radial velocity is always positive, breakdown must eventually occur. The analysis also shows, that the outer portion of the vortex cannot support a pressure
The problem of vortex breakdown
381
gradient along the axis, if the radial distribution of the azimuthal velocity c o m p o n e n t is equal to that of a potential vortex; that a positive pressure gradient Op/Ox(x, 0) results, if the radial decay of the azimuthal velocity components is less steep than that of the potential vortex for positive radial velocity components, and that an increase of the initial axial velocity c o m p o n e n t delays breakdown as does a decrease of the m a x i m u m azimuthal velocity component. The results given in this analysis a valid for large Reynolds numbers with small core radii. Extension to large core radii should be possible.
REFERENCES I. S. Leibovich, The Structure of Vortex Breakdown. Ann. Rev. Fluid Mech. 10, 221-246 (1978). 2. M. G. Hall, Vortex Breakdown. Ann. Rev. FluidMech. 4, 195-218 (1972). 3. X. Shi and E. Krause, "Numerische Untersuchung des Aufplatzens eines Wirbels." Paper presented at the GAMM Tagung 1983, Band 64, T230-232. 4. T. Sarphaya: On Stationary and Traveling Vortex Breakdown. J. Fluid Mech. 45, 545-559 {1971). 5. R. Staufenbiel: Private Communication (1982).