AIAA 91-1658 Numerical Evolution of Elliptical 3-D Vortex Arrays in Inviscid Fluid Xiong John He, and Jacques Lewalle, Syracuse University, Syracuse, NY 13244 and Jackson Herring, MMM, NCAR, Boulder, CO 80307-3000

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.- 2nd Fluid Dynamics, Plasma Dynamcs

& Lasers Conference June 24-26, 1991 / Honolulu, Hawaii I-

For permission to copy or republish, contact the Amerlcan Institute of Aeronautics and Astronautics 370 CEnfant Promenade, S.W., Washlngton, D.C. 20024

AIM-91-1 658-CP NUMERICAL EVOLUTION OF ELLIPTICAL 3-D VORTEX ARRAYS IN

INVISCID FLUID

Xiong John He*, and Jacques Lewalle, Mech. Engineering Dept., Syracuse University, Syracuse, NY 13244

and Jackson Herring, MMM, NCAR, Boulder, CO 80307-3000

A b s t r a c t

A numer ica l evolu t ion of vortex arrays in an inviscid fluid was performed by thc Galerkin-Fourier spectral method. The initial f lows include Deissler vortex and Taylor -Green vortex. The init ial length scales along different directions are cqual (c i rcular) and unequal (ell iptical) , 3nd thc rcsults are compared. The Deissler vortex flows have much stronger vortex interaction than the T-G vortex flows. The clliptization enhances the nonlinear inter- actions obviously in the doughnut T-G flow, but the enhancement is weak in the plane T-G flow and the Dcissler flow. The results support the concept that a main source of turbulence i s the nonl inear interact ion between neighboring vortex segmcnts ori- ented di’fferently.

N o m e n c l a t u r e

C1, C2 and C3 symbols used in Eqs. 15, 16,

ka wavenumber vector . K cl l ipt izat ion parameter . Ga turbulence o r vortex interaction

P a p and Pap7 direct ional projectors,

Sap st rain-rate tensor . Ua velocity vector in physical space. V a velocity vector i n wavenumber space. t t ime. X a ??ace coordinates. a,!$ and y used as subscripts for

directional components . EaPy unit skew-symmetric tensor. v kinetic viscosity. Ra vorticity vector.

and 19, 20.

p a r a m e t e r .

defined in Eqs. 10 and 1 1 .

1 . Introduct’ IOU

Turbulent f low is a common and important phenomenon in nature and in engineering problems. It is well known that the turbulence is closely related to thc instability of laminar flow and the nonlin- car interactions distributed in the flow. When a vortex is stretched along its rota- tion axis, the local vorticity is increased, and vice versa. A single plane-rotating two-dimensional vortex. such as a potential eddy (except the central part) is stable. The rotations in a coffee-cup or a kitchen s ink are well-known examples of the stable 2-D cddies (with’’ a weak viscosity dissipation). Based on the calculated growth of instabil- ity with the eccentr ic i ty of e l l ipt ical s t reaml ines , B a y l y l and Pierrehumberth suggested that the short-wave instability of two-dimensional ell iptical vortices i s an- o ther general mechanism of tu rbulence g e n e r a t i o n .

The evolution of vortex arrays is an cf- fective model of the path to turbulence. and i t has been studied by a number of authors using various methods. For example. the Taylor-Green vortex flow is a classical prob- lem studied with finite difference, t ime- ser ious expansion, and spectral methods ( T a y l o r and G r e e n 7 , Brachet et D e i s s l e r 3 calculated another turbulent solu- tion of the Navier-Stokes cquations from a three-dimensional initial vortex array. The 3-D initial vortex flow has a complex chaotic pattern of streamlines and the flow dr:.el- opment exposes some turbulence character- istics. This paper extends these results 10 the non-circular vort ices , and shows a fas ter breakdown for arrays of cll iptical vo r t i ce s .

*Member of AIAA. Present address: Physics Dept., Rutgers University, Newark, NJ 07102.

2

2 Governing Eawt ions and Soect ral Method

Galerkin-Fourier spectral method has become a power tool with the rapid develop- ment of computational technology in the last decades. The spectral method is par- t icularly powerful for solving nonlinear evolut ion problems when the boundary conditiom are of secondary importance and can be modeled as periodic in space. All typical turbulence flows are characterized by high Reynolds numbers, small viscosity, and dominant inertial effect. The dynamics of the turbulence cascade is based on the inertial property of the fluid and, except a narrow region near a solid boundary, it is weakly dependent on the boundary condi- tions. Therefore , a per iodic boundary condition is suitable for the study.

Following the standard algorithm of the Galerkin-Fourier spectral method, we in t roduce the ve loc i ty image in the wavenumber space by 3-D Fourier trans- f o r m a t i o n

The Navier-Stokes equations can be written as:

Va(k19k2,k3;t) = F T ( U ~ ( X ~ , X Z . X ~ ; ~ ) ) . ( 1 )

-_ - Ga - v k2 V, ava at

where the continuity relation

and the identity relation aVa/axa = 0 ( 3 )

have been used. For flows in an inviscid f luid, the gove rn ing cquat ion becomes auite s i m d e :

= L G a at (5)

The variable Ga i s called the turbulence interaction parameter in an ideal uniform fluid and i t is defined bv

There arc several equivalent expressions for the Ga , iuch as

G a = Pap F U E ~ y q Uy Qq) ( 7 ) (8) ( 9 )

= Pap FT( Uy Spy ) = Papy JT E P ~ u y Uq) .

where Pap and Papy are directional pro- j ec to r s :

Pap = ( Sap - k a kp /k2 ) . (10) v

( 1 1 ) " I P a b y = 5 ( Pay k p + P a p ky ) .

The variables inside the F T ' s brackets in the above three expressions of G , (Eqs. 7 to 9) are products of velocity and its space der iva t ives . T h e th ree products a re unequal each other, but their products with corresponding projectors are equal. The form of G , used most frequently is the Eq. 7, because it includes only a cross product of velocity and vorticity without tensor. The projectors dis t r ibute the nonlinear inter- action between neighboring fluid particlcs onto each direction in the whole space. Thc distributed interaction determines the flow evolution according to Eq. 5.

I t is well known that the Bernoulli function i s a constant inside an inviscid steady irrotational flow. I t is clear also from Eq. 7 that G a is zero everywhere in a steady irrotational f low. Thereforc the evolution of an unsteady rotational flow (the left hand side of Eq. 5) relates to the variation of the Bernoulli function as i t relates to the €low's rotationality. To study - the turbulence mechanism, we focus on Ga. Eq. 7 shows that an inviscid flow is steady if the velocity vector and the vorticity vcctor r ema in para l le l t h roughou t t h e f i e ld (Beltrami condition). In such a steady ro- tating flow, the velocity variation is totally balanced by the dynamic pressure distri- bution. A famous example of the Bcltrami flows is the ABC flow:

U a = (A sin xz + C cos x3. B sin x3 + A c o s x l , B s i n x l + C c o s x 2 ) ( 1 2 )

The ABC flow has chaotic streamlines and i t is steady, as the velocity and vorticity arc paral le l everywhere (see, for cxamplc , Domhre et al.5).

The above three expressions for the parameter G a derive three explanations for the turbulence interaction. They are bascd on (1) the interaction between neighbor- ing vortex segments; (2) the interaction between neighboring strain-rate segments; _~, and (3) the gradient of velocity cross prod- ucts, respectively. Considering the direc- , tional project, all the three explanations are trans-directional and equivalent.

3

v

W The first explanation based on the vortex interaction is drawn in Fig. 1, where a main vortcx is selected in the x3 direction. A single vortex without neighboring vor- tices is steady, although the velocity is per- pendicular to the vorticity. The kinetic en- ergy distribution in the single vortex is balanced by the dynamic prcssurc as the Bernoulli theorem requires. The balance rcflccted in the governing equation is the zcro products of e a p r U r Q p with thc direc- tional projector, Pup.

Figurc I . The interaction between neigh- boring vortex segments oriented differ. e n t l y .

v

W

When a non-parallel vortex (Q2 in Fig. I ) exists in thc neighborhood of the above Q 3 vortex, the variation of the velocity vector through the vortex is not entirely balanced by a dynamic pressure variation. Thus a non-vanishing G a (a t least the com- ponent G3 for the case shown in Fig. 1) is created. Generally, the non-zero Ga de- tcrmines the flow evolution, including the development toward turbulence. Therefore, the Ga is called the turbzlencc interaction parameter o r the vortex interaction pa- rametcr. The explanation shows that the interaction. G a , is related to the dynamic pressure rcsponse and i t may be in different length scales and along different d i r e c t i o n s .

The above governing cquation system (Eqs. 5 and 7) was solved with the initial conditions of Taylor-Green vortex array, Deissler vortex array, and their elliptical variations. The time discretization was per- formed by a s torage-saving third-ordcr Runge-Kutta method (Wil l iamson8) . Thc number of mesh points was up to 6 4 x 6 4 ~ 6 4 in a periodic cube with periodic boundary conditions. The program was run on the Cray at National Center for Atmospheric Research, a VAX at Syracuse University, and the IBM 3 0 9 0 a t Cornell Nat iona l Supercomputer Faci l i t ics .

3. The Initial Circular and Elliptical Vortex Arravs Used in the Simulation

The initial condition of the circular T -

U 1 = sin(x1) cos(x2) cos(x3)

U3 = 0.

G vortex used in the simulation is

U 2 = - cos(x1) sin(x2) cos(x3) (14 )

Based on the analysis given in Brachet et a1.2, this selection of initial conditions is not less general than other forms. The main rotation axis in the initial T-G vortex is along the x3 direction and the streamlines are near-circular closed curves in the x 1 - x ~ planes. In fact. the initial T-G vortex is a two-dimens iona l f low per iodica l ly dis- tributed along its rotating axis. The bound- aries of each rotating cubic element in the T-G vortex are motionless (for cxample. it is kept that Ul=U2=U3=0 at x3=(n+1/2)fc) and thc flow structure is s imilar to beads on a n e c k l a c e .

Ano the r ini t ia l vortex array with simple Fourier spectrum used i n the simu- lation i s the three-dimensional Deissler vortex. Deiss ler3 suggested the initial condition and solved i ts development in a low viscosity fluid. The initial condition is

u 1 = 2 c 1 +a+ c 3 u 2 = c 1 + 2 c2 + c 3 u 3 =c1+ c 2 + 2 c 3

c 1 = cos(- XI + x2 + x3 ) c 2 = cos( X I - x2 + x3 ) c 3 = cos( X l + x2 - x3

( 1 5 )

w h e r e

( 1 6 )

4

The Deisslcr vortex has complex 3-D stream- lines which can not easily be drawn on a plane. The Deissler vortex is symmetric along the three axes, though i t is not i so t rop ic .

The init ial e l l ipt ical vortex arrays used in the study include both the elliptical T-G vortices and the clliptical Deisslcr vor- tcx. Considering the orientations between the clliptization and the rotation, the initial elliptical T-G vortex includes two cases: the plane clliptical T-G vortex

U 1 = (1K) sin(Kx1) cos(x2) cos(x3)

U3 = 0. and the doughnut clliptical T-G vortex

U 1 = sin(xl) cos(x2) cos(Kx3) U2 = - cos(x1) sin(x2) cos(Kx3) U3 = 0.

U2 = - cos(Kx1) sin(x2) cos(x3) (17)

( 1 8 )

The two elliptical T-G vortex arrays are shown in Fig. 2. I t is clear that in the plane clliptical T-G vortex, the shape of stream

W

Table I. A Comparison of the Studied Initial Vorticcs

T-G Vortices Deisslcr Vortices Circular Elliptical Circular Ellip.

Plane Doughnut K=l K=2 K=2 K=l K=2

Max. velocity

Max. relative

Max. G , in

component 1 1 1 4 4

vorticity 2 2.5 2 1.5 2.25

Fourier P.03125 4*.03123 4*.05 4*1.5 4*1.5 components +4*.01246 +2*.3003

+2* ,001 6 c . . .

lines has been changed to ellipses on rota- tion planes, bur the distance bctwccn the planes are kept. The largest initial vortic- ity in the plane elliptical T-G flow is also increased (Table I ) . In the opposite, the doughnut elliptical T-G vortex have circu- lar streamlines, but, thc distance between these planes are shorten. The largest com- ponent of vorticity (Q 3 ) in the initial doughnut T-G is the same as the circular T-G vortex, cvcn the minor vorticitics (Q 1 and Q 2 ) i n perpendicular directions are en- hanced. In addition, the distances of the neighboring vortex segments oricnted d i f - ferently arc shorted in doughnut T-G flow, because the distance between the rotation planes is shorted.

4

The elliptical initial Deisslcr vortex is: u1= (2 c 1 + c 2 + c 3 )/K u 2 = c1+ 2 c2 + c 3 (19) u3 = CI + c2 + 2 c 3 ,

C2 = COS( Kxl - ~2 + ~3 )

w h e r e C1 = COS(- Kxl + x2 + x3 )

C3 = COS( Kxl + x:! - x3 ). (20)

The elliptical parameter, K. was taken equal to 2 in most calculation. The clliptization does not change the flow structure, thoudl .- I

i t causes a small decrease of the maximum - speed and a minor increase of somc vortic- nut (low-half) elliptical Taylor-Green

. . - -. . - - - i tv comuonents. Therefore the relativc d

Figure 2. The plane (up-half) and dough-

V U I l l t i G S . rotation strength is enhanced a little by the elliptization in Deissler flows.

5

4. The Evolution of Circular Vortex Arravs in a U niform Fluid

'The computing program runs well before the offspr ing subsca lc s t ructures mature and inherit a large part of the total kinetic cnergy. For plane rotation vortex array as the T-G flow, the initial main vor- tex may complete a few revolutions before losing its initial kinetic energy (see Fig. 3). B u t i n a three-dimensional vortex array as the Deissler flow, the initial vortex loses a large part of its initial kinetic energy and the flow structure exhibits turbulent char- acterist ics before the initial vortex com- pletes a rcvolution. I f calculations are continued beyond this point without addi- iion of modes for better rcsolution. the data lluctuate spuriously, the total kinetic en- crgy in the flow is not conscrved and the calculation loses physical meaning.

1.0

. I \ -\

0 . 8 -

0 . 0 -

I . ( -

The results given in Fig. 3 shows that the Deissler 3-D vortex has a stronger vor- tex interaction, generates subscale vortex structures faster than the T-G vortex flow. The Deissler flow develops towards turbu- lence faster than the T-G vortex flows, although both the flows initially have the same relative strengths of rotation.

For turbulence mechanism, the local maximum propert ies are more important than the averaged properties, because the local proper t ies de te rmine the internal processes inside the turbulence. Table 1 compares the maximum components of velocity, vorticity, and vortex interaction parameter in the above two initial flows and their elliptical variations. The first row lists the maximum component of the velocity vector. The other rows list the re la t ive s t r e n g t h s of f low p rope r t i e s (divided by corresponding maximum ve- locities). The second row shows that the relative rotation strengths are of the same level in all the studied flows, including the elliptical cases. Similar calculation (data not listed here) shows that the relative strengths of strain-rates for al l the studied flows are of the same magnitude.

The third row in Table 1 lists the largest directional component of the vortex interaction parameter , G a , in the Fourier component form. For example, the first element in the third row is 8*0.03125 which means that the largest directional compo- nents of G a has eight Fourier components in magnitude of 0.03125 (with different wavenumbers). The next element i n the row shows that the largest Ga in the plane elliwtical T-G vortex has eight Fourier com- .., ponents: four with magnitude of 0.03125 and four with 0.01246. Calculation shows that all these components (except some

are in a larger wavenumber region, there- Dinensionlei5 l i n t fore the G a create microscopic structures

1<1111. I-G

components in the elliptical Deissler flow) 0 . 0

1.0 1 , s 2.0 2.5 0.0 0 . 5

a n d t h e n g e n e r a t e s t u r b u l e n c e . Figure 3. A comparison of the decay of ih3 Considering the difference i n the magni-

largest vortices and the development of tudes of the Ga. i t is not surprising that-the subscale StI'UctureS i n circular Deissler flow develops toward turbulence Green and Deissler flows. much faster than the T-G flows.

v

W

6

,xel . l ."e

1 . 2 0

1 I S

1 . 1 0

L.05

1 . 0 0

a 9 5

1

W

W

r i s , , c k L < " t "

~ . . ~ ........

5 . Effects o f ElliDtization on Tavlor-Green Vortex Flow

to I t , l n i r l d l I d j Y C ,

'+,I ~ Q n , t - o ~

-

.

_ - _ _ _ SA p > a n s 7-c. x - 2

-. -. dougnnvt ,Z.,X'- T.G. a-2 - -- - -- .

\ 'L,

( 1 . 1 . I . ,

" \.,'> 0 1 0 1 L I m c _, \ 0.1 0.2

A s imi la r simulation was performed f o r bo th e l l i p t i ca l Tay lo r -Green and Deissler vortex arrays. The elliptization e n h a n c e s t h e vo r t ex in t e rac t ions and accelerates the f low development toward turbulence in the doughnut elliptical T-G vortex as shown in Figs. 4 and 5. The upper half of Fig. 4 shows the decrease of the main vortex kinetic energy and the lower half shows the increase of kinetic energy i n the first overtone harmonic component (at larger wavenumbers). The curves in Fig. 4 show also that the elliptization has a r a t h e r s m a l l e n h a n c e m e n t and e v e n reduction on the turbulence interaction in the plane T-G flow.

Gcncral ly , the microscopic structures in turbulence have small size. large veloc- i t y , large vorticity, and near-isotropic dis- tribution. Therefore a growth of the maxi- mum vorticity (and velocity) components and an evolution of field toward isotopic are consis tent with the development toward turbulence. The lower half of Fig. 5 com- pares the decays of the main vortices in the T-G flows, and shows that the evolution is accelerated in the doughnut T -G flow and delayed a little in the plane T-G flow. The upper of Fig. 5 shows the same effect of el- l ipt izat ion through the growth of the smaller vortices. Even there arc two ben- cfit factors ( the increase of Ihe maximum vorticity component and the shape change of the streamlines) in the plane T-G flow, the rcsults expose that the cll iptization affects the flow evolution in plane T-G flow is much weaker than i t in doughnut T-G flow. A possible explanation of the result is that the distance and the rclative orienta- tion of the interacting segments of neigh- boring vortices are important in determin- ing the nonlinear turbulence interaction.

The circular T-G vortex array has a main vorticity along x3 and weak vorticities along X I and x2. Fig.1 shows thc interaction between R 3 and in the circular T-G flow. The doughnut elliptical T-G vortex array

I

7

v

increases the weak vorticities R 2 and R 1 and keeps the R 3 . The elliptization in the doughnut T-G vortex also shortens the dis- tances between the main vorticity R 3 and the weak vorticities R1 and R 2 . Therefore t h e vortex interaction is increased obvi- ously as the simulation results.

v

In plane T-G vortex, a main effect of the ell iptization is to increase the main vorticity R 3 . The distance between the vort ices or ientated diffcrcnt ly docs not changed, Therefore, the weak effect of elliptization in the plane T-G vortex shows that t hc relative orientation o f . interacting vort ices is important i n creating turbu- lence interaction.

6. E f f m s o f Elliptization Q n Deissler Vortex

0 2 0 1 t,ms Calculat ion resul ts show ' tha t the 0 1 - .. elliptization in the Deissler flow increases the initial G a (Table l ) , where the simula- tion for the flow evolution shows that the carly stage of the development to turbu- lence is a little slowed by the elliptization in the Dcisslcr flows. This is because the non- l inear interact ion creates kinetic energy transfers toward both larger and smaller vortices (i.c., some the early Gu components have wavenumbers less than ( K 2 + l + l ) 0 . 5 and s o m e c o m p o n e n t s h a v e w a v e n u m b e r s larger than it). After a quite short time (in ou r simulations, above 0.1). t he energy t r ans fe r t oward s m a l l e r w a v e n u m b e r s decreases to near zero and the energy transfer toward larger wavenumbers con- trols the flow development. Fig. 6 shows the effect of elliptization on the rotation prop- crty development in Deissler flows.

v

The circular Deissler vortex is a three- dimensional flow. The neighboring vortex segments oriented differently (as the case shown in Fig. 1) are distributed throughout in the Deissler flow. So that the segments - interact strongly inside the flow to create and enhance the offspring structures. The

v elliptization changes the relative scale dis- t r ibut ion among direct ions, and weakly affects the rotation strengths. The general

Figure 6. The evolution of vorticity com- ponent maximum of elliptical Deissler f lows.

3-D characteristics are kept in the consid- ering elliptization for the Deissler flows. Therefore, the elliptization has only a weak effect on the flow evolution toward turbu- lence as the simulation results exposed. In other words, the calculation results support the concept that a main mechanism of tur- bulence interaction i s t he vortex interac- tion between neighboring vortex segments orientated different ly .

c o n c l u s i o n

Numerical simulations for the evolu- tions of vortex arrays are performed with the Galerkin-Fourier spectral method. The ini t ia l cond i t ions inc lude Tay lo r -Green vortex, Dcissler vortex and their elliptical variations. All the initial T-G vortex flows have closed streamlines on parallel planes, and the initial Deissler vortices have 3-D streamlines. The results shows that the tur- bulence interaction i s enhanced by ellipti- zation in the doughnut T-G flow, but the

8

effect is weak in the plane T-G flow. All the Deiss ler f lows have much stronger vortex interaction than the T-G flows. The ellip- ticity weakly a f fec ts t he development in Deissler flows also. The above results can be explained by the mechanism of turbu- lence interaction which is created by the in t e rac t ion b e t w e e n ne ighbor ing vor tex segments or ien ted different ly .

A c k n o w l e d e e m e n &

T h e study was supported by Syracuse University and the code was run on com- puters in Syracuse University. NCAR, and CNSF. The authors would like to thank Dr. Daniel Murnick for comments and advice in the manuscr ip t preparat ion.

References

I . Bayly, B., 1986: Three-dimensional inslability of

2. Brachet. M. E., D. 1. Meiron. S. A,, Orszag, B. G. elliptical flow. Phys. Rev. Let., 57, 2160-2163.

Nickel, R. H. Morf, and U. Frisch, 1986: Small scale structures of the Taylor-Green vortex. J. Fluid Mech., 130, 411-452.

of fluid motion. Rev. Mod. Phys., 56, 223-254.

processes in 2-D and 3-D vortices in numerical simulation, AIM Paper 89-0610.

5. Dombre, T., U. Frisch, J. M. Green, M. Henon, A. Mehr, and A. M. Soward, 1986: Chaotic streamlines in the ABC flows. J . Fluid Mech., 167, 353-391.

6. Pierrehumken, R.. 1986: Universal short-wave insrability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Let., 57, 2157-2159.

7. Taylor, G. I. and A. E. Green, 1937: Mechanism of the pnuluc!ion of smd! eddies from large ones, Proceedinns of the Roval Socieri of London.

3. Deissler, R.. 1984: Turbulence solutions of the equation

4. He, J. X., and J. Hening, 1989: Turbulence dissipation