Computation of compressible quasi-axisymmetric slender vortex flow and breakdown

164

Computer Physics Communications 65 (1991) 164 172 North-Holland

Computation of compressible quasi-axisymmetric slender vortex flow and breakdown Osama A. Kandil ~ and Hamdy

A. Kandil 2

Department of Mechanical Engineering and Mechanics. Old Dominion Uni~,ersi(v, Norfoll,, VA 23529-0247, USA

Analysis and computation of steady, compressible, quasi-axisymmetric flow of an isolated, slender vortex are considered. The compressible Navier-Stokes equations are reduced to a simpler set by using the slenderness and quasi-axisymmetry assumptions. The resulting set along with a compatibility equation are transformed from the diverging physical domain to tl rectangular computational domain. Solving for a compatible set of initial profiles and specifying a compatible set of boundary conditions, the equations are solved using a type-differencing scheme. Vortex breakdown locations are detected by the failure of the scheme to converge. Computational examples include isolated vortex flows at different Mach numbers, external axial-pressure gradients and swirl ratios. Excellent agreement is shown for a bench-mark case between the computed results using the slender vortex equations and those of a full Navier Stokes solver.

1. I n t r o d u c t i o n

The p h e n o m e n o n of vortex b r e a k d o w n or bursting was o b s e r v e d in the water v a p o r cond e n s a t i o n trails along the leading-edge vortex cores of a gothic wing. Two forms of the l e a d i n g - e d g e vortex b r e a k d o w n , a b u b b l e type a n d a spiral type, have been d o c u m e n t e d e x p e r i m e n t a l l y [1]. The b u b b l e type shows an almost a x i s y m m e t r i c s u d d e n swelling of the core into a b u b b l e , a n d the spiral type shows an a s y m m e t r i c spiral filament followed by a r a p i d l y s p r e a d i n g t u r b u l e n t flow. Both types are characterized b y an axial stagnation point a n d a limited region of reversed axial flow. M u c h of our k n o w l e d g e of vortex b r e a k d o w n has been o b t a i n e d from e x p e r i m e n t a l studies in tubes where b o t h types of b r e a k d o w n a n d o t h e r types as well were g e n e r a t e d [2-4]. The m a j o r effort of n u m e r i c a l s i m u l a t i o n of vortex b r e a k d o w n flows has been focused on incompressible, q u a s i - a x i s y m m e t r i c isolated vortices. G r a b o w s k i a n d Berger [5] used the incompressible, q u a s i - a x i s y m m e t r i c N a v i e r - S t o k e s equations. H a -

i Professor and Eminent Scholar. Graduate Research Assistant.

fez et. al [6] solved the incompressible, steady, q u a s i - a x i s y m m e t r i c Euler a n d N a v i e r - S t o k e s e q u a t i o n s using the stream f u n c t i o n - v o r t i c i t y form u l a t i o n a n d p r e d i c t e d vortex b r e a k d o w n flows similar to those of G a r b o w s k i a n d Berger. Spall, G a t s k i a n d G r o s c h [7] used the vorticity velocity f o r m u l a t i o n to solve the three-dimensional, incompressible, u n s t e a d y N a v i e r Stokes equations. F l o w s a r o u n d highly swept wings a n d slender w i n g - b o d y c o n f i g u r a t i o n s at transsonic a n d supersonic speeds and at m o d e r a t e to high angles of a t t a c k are c h a r a c t e r i z e d by vortical regions and shock waves, which interact with each other. Other a p p l i c a t i o n s which e n c o u n t e r v o r t e x - s h o c k intera c t i o n include a s u p e r s o n i c inlet ingesting a vortex and injection into a s u p e r s o n i c c o m b u s t o r to enh a n c e the mixing process, see Delery et. al [8] and Metwally, Settles a n d H o r s t m a n [9]. These p r o b lems a n d others call for d e v e l o p i n g c o m p u t a t i o n a l schemes to predict, s t u d y a n d control compressible vortex flows a n d their interaction with shock waves. U n f o r t u n a t e l y , the literature lacks this type of analysis with the exception of the p r e l i m i n a r y work of Liu, K r a u s e a n d M e n n e [10] a n d C o p e n i n g a n d A n d e r s o n [11]. In this p a p e r , the steady, c o m p r e s s i b l e N a v i e r Stokes e q u a t i o n s are simplified using the quasi-

0010-4655/91/$03.50 ,:'~1991 - Elsevier Science Publishers B.V. (North-Holland)

O.A. Kandil and H.A. Kandil / Compressible quasi-axisymmetric slender vortex flow axisymmetry and slenderness assumptions. A compatibility equation [10] has been used and the governing equations are transformed to a rectangular computational domain by using a LeveyLee-type transformation. A compatible set of initial conditions and boundary conditions is obtained and the problem is solved using a type-differencing scheme. The numerical results show the effects of compressibility, external axial pressure gradients and the swirl ratio on the vortex breakdown location. A bench-mark flow case has been solved using these equations and the full NavierStokes equations. The results are in excellent agreement with each other.

165

and f(p) is a function relating the density integral at any axial station to that at the initial station. It is equal to 1 for incompressible flow. The subscript e refers to external conditions and the subscript i refers to initial location. The governing equations become ?,

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where

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0

XU

and

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1 ap p a~

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2. Highlights of the formulation and computational scheme Starting with the steady, compressible NavierStokes equations which are expressed in the cylindrical coordinates (£, F and q,), assuming the isolated vortex flow to be slender [~/1= 0 ( 1 / R f ~ ) , 6/Uoo = O (1/Rye-), where l is a characteristic length, 6 the radial velocity, U~ the freestream velocity and Re the freestream Reynolds number] and quasi-axisymmetric [ 8 / 8 ~ ) = 0], and performing an order-of-magnitude analysis, the equations are reduced to a compressible, quasi-axisymmetric, boundary-layer-like set. The dimensionless flow variables P, P, u, v, w, T and g are non-dimensionalized by O~, o~a 2, a~, a~/Cp and g~ for the density, pressure, velocity, temperature and viscosity, respectively, where Cp is the specific heat at constant pressure. Next, we introduce a Levey-Lee-type transformation which is given by x ~ = fo Pel'te d x ,

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The other compatible initial conditions are obtained from a compatibility equation and eqs. (5)

O.A. Kandil and H.A. Kandil / Compressible quasi-axisymmetrie slender vortex flow

166

and (8). At the vortex axis, ~ = 0, we specify Ou

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O.A. Kandil and H.A. Kandil / Compressible quasi-axisymmetric slender vortex flow

dure consists of two parts. In the first part a compatible set of initial profiles are obtained at = ~i and in the second part we use eqs. (4)-(8) and the compatibility equation to obtain p, u, w, p, T and V (or v).

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3. Numerical examples In the present numerical examples, the outer edge of the vortex, ~ , is taken as 10, and 1000 grid points are used and hence A ~ = 0.01. The

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O.A. Kandil and H.A. Kandil / Compressible quasi-axisymmetriv slender vortex flow

168

results are shown for two Mach n u m b e r s : M = 0.5 a n d 0.75. The step size in the axial direction is 0.02 for M = 0.5 a n d 0.04 for M = 0.75. F o r each M a c h - n u m b e r case, we solve for two external axial pressure gradients; (Op/OX)e = 0.125 a n d 0.25 a n d two swirl ratios; fl = (w/u)~=~ = 0.2 a n d 0.4. The initial profiles for u i, w~ a n d Ti are u i = constant, wi = f l u i r ( 2 - r 2) for r < l a n d wi = f l u i / r for

r > 1 a n d T i = 2.5, respectively. Fig. 1 shows MSF, Ua, Pa a n d Ta which are referred to by curves A, B, C a n d D; respectively. The results show that the b r e a k d o w n length is more than d o u b l e d when the Mach n u m b e r increases from 0.5 to 0.75. They also show that while the outer b o u n d a r y c o n t i n u ously increases for M = 0.5, it initially decreases a n d then increases for M = 0.75: see the A curves.

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Fig. 2. F l o w profiles f o r s l e n d e r q u a s i - a x i s y m m e t r i c f l o w s at M = 0.5 a n d 0.75, fl = 0.4, ( d p / d x ) e = 0.25.

- E.4

O.A. Kandil and H.A. Kandil / Compressible quasi-axisymmetric slender vortex flow

The adverse pressure gradient at the vortex axis decreases faster for M = 0.75 than for M = 0.5. The results also show that the external axial pressure gradient is a dominant parameter on the breakdown length. As the external axial pressure gradient is doubled, the breakdown length substantially decreases. Doubling the swirl ratio slightly decreases the breakdown length.

Fig. 2 shows the profiles of u, w, p and across r at axial stations until the breakdown location for M = 0.5 and 0.75 for the cases of (dp/dx)~ = 0.25 and /3 = 0.4. The initial profiles are indicated by the number 1 and the next shown station is indicated by 3. At M = 0.75, it is noticed that the pressure and density gradients in the axial direction decrease faster than those at M = 0.5.

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O.A. Kandil and H.A. Kandil / Compressible quasi-axi3ymmetric slender vortex flow

170

plane is used. The curves are labeled by the capital letters A, B, ... etc. C o m p a r i n g the curves of the two sets, a remarkable agreement is seen. It is concluded from the given numerical examples that increasing the flow M a c h n u m b e r has a favorable effect on the vortex breakdown location. The external axial pressure gradient is a d o m i n a n t parameter on the vortex breakdown. Its effect

The profiles show that the viscous diffusion at M = 0.75 is larger than that at M = 0.5. Fig. 3 shows the profiles of u, w, v and p which has been c o m p u t e d by the present m e t h o d and by an upwind N a v i e r - S t o k e s solver for the case of M = 0.5, /3 = 0.6 and (dp/dx)e = 0. For the N a v i e r - S t o k e s solver a rectangular grid of 100 × 51 × 51 in the axial direction and cross-flow

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/3 = 0.6, ( d p / d x ) ¢ = 0.0.

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~J

--"

ci

i"4I

~ •4

~..a I.2 '

,

•1 ~.~ 62 i

i

Vertical H=E.5 b e t a = E • 6

Velocity , w d x = E . 2 F u l l N-S

.63

,

•64

.65

•~

•67

.68

•69

Pressure . p H-E.5 bst&-E,6 dx=~.2 Full

N-S

.7g

•71

.72

Fig. 3 (continued).

decreases as the Mach number is increased. Comparison of the present results with the full NavierStokes results gives a strong confidence in the present analysis. The present formulation and results are used to generate compatible initial profiles for the full Navier-Stokes solutions, and to provide data for breakdown-potential cases for accurate computations using the full NavierStokes equations. The fuU Navier-Stokes equa-

tions are currently applied to these cases, so that we can solve for the flow in the breakdown region.

Acknowledgement

This research work is supported by the NASA Langley Research Center under Grant No. NAG1-994.

172

O.A. Kandil and H.A. Kandil / Compressible quasi-axisymmetric slender vortex flow

References [1] N.C. Lambourne and D.W. Bryer, Bursting of leading-edge vortices: Some observations and discussion of the phenomenon, Aeronautical Research Council, R&M 3282 (1961). [2] T. Sarpkaya, Vortex breakdown in swirling conical flows, AIAA J. 9 (1971) 1791. [3] S. Leibovich, Vortex stability and breakdown survey and extension, AIAA J. 23 (1984) 1194. [4] M.P. Escudier and N. Zender, Vortex flow regimes, J. Fluid Mech. 115 (1982) 105. [5] W.J. Grabowski and S.A. Berger, Solutions of the Navier-Stokes equations for vortex breakdown, J. Fluid Mech. 75 (1976) 525.

[6] M. Hafez, G. Kuruvila and M.D. Salas, Numerical study of vortex breakdown, J. Appl. Num. 2 (1987) 291. [7] R.E. Spall, T. Gatski and C.E. Grosch, A criterion for vortex breakdown, ICASE Report 87-3 (January 1987). [8] J. Delery, E. Horowitz, O. Leuchter and J.L. Solignac, Fundamental studies of vortex flows, Rech. Aerosp. No. (1984) 1. [9] O. Metwally, G. Settles and C. Horstman, An experimental study of shock wave/vortex interaction, AIAA Paper 89-0082 (January 1989). [10] C.H. Liu, E. Krause and S. Menne, Admissible upstream conditions for slender compressible vortices, AIAA Paper 86-1093 (July 1986). [11] G. Copening and J. Anderson, Numerical solutions to three-dimensional shock/vortex interaction at hypersonic speeds, AIAA Paper 89-0674 (January 1989).