I AIAA-89-0122

U N I F I E D FORMULATION FOR INCOMPRESSIBLE FLOWS

D. Pan and S. Chakravarthy Rockwell International Science Center Thousand Oaks, CA 91 360

27th Aerospace Sciences Meeting January 9-12, 1989/Reno, Nevada

L

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics (3 370 L’Enfant Promenade, S.W., Washington, D.C. 20024

Unified Formulat ion for Incompressible Flows

Dartzi Pant and Sukumar R. Chakravarthyr Science Center, Rockwell International

Thousand Oaks, California, USA

Abst rac t A unified computational method originally devel-

oped for compressible flows has been applied to solve the artificial compressibility equations for 2D and 3D incompressible flows. The method is a finite-volume implementation of high accuracy upwind schemes em- bedded in a multi-zone, structured-grid book-keeping framework. The unified formulation permits both steady- state and timeaccurate computations using either time- marching or space-marching solution techniques. The continuity equation was modified from its conventional formulation to allow grid motions. A simple, general and nonsingular eigensystem was derived for 3D flux Jacobians. The iterative process derived from New- ton’s procedure was modified to satisfy the divergence- free requirement at every time level. The relationship between artificial compressibility and artificial dissipa- tion was discussed. It was shown that the momentum equations are contaminated by artificial compressibil- ity in upwind schemes. This represents a potential limitation on the use of upwind schemes based on ar- tificial compressibility, although it is seldom reached in practice.

1. Introduction By adding an artificial time-derivative term of pres-

sure to the continuity equation, the incompressible flow equations are rendered hyperbolic and all the well- developed compressible flow algorithms can be read- ily applied with little modification. This attractive idea of artificial compressibility was first suggested by Chorin.’ Later i t was implemented by Steger and Kutler’ and Kwak et a13 for steady-state problems using cen- tral difference schemes. Chakravarthy4 used SCM u p wind method to solve incompressible Navier-Stokes equa- tions in Cylindrical system in inertial and rotating frames. Only recently timeaccurate computation based on artificial compressibility was attempted by Merkle and Athavale,‘ and a flux-differnece upwind scheme was implemented by Hartwich and H s u . ~

This paper is the result of an effort to apply a uni- fied solution alg~rithm’,~ originally developed for com- pressible flows to solve the artificial compressibility equations for incompressible flows. The unified algm rithm is based on a finitevolume implementation of high accuracy (up to third-order) flux-difference up-

-

was well-proven for compressible flows ranging from subsonic t o hypersonic flow regimes. Reference 8 has a detailed outline of the capabilities of this algorithm.

The addition of artificial compressibility changes the elliptic-parabolic nature of the incompressible Navier- Stokes equations t o hyperbolic-parabolic. It enables the use of the above mentioned algorithmin astraight- forward manner. However, because of the divergence- free requirement for incompressible flows, there are some important modifications that must be implemented. First, in order to allow general grid motions, the conti- nuity equation must be modified from its conventional formulation and the eigensystem of the flux Jacobians must be changed accordingly. Second, the implicit it- erative process based on Newton’s procedure for ad- vancing the solution needs to be modified to satisfy the divergence-free requirement in a time-accurate fashion. To improve the efficiency further, a simple, general and nonsingular eigensystem for 3D flux Jacobians was de- rived which requires metries information in the con- sidered direction only.

Since the hyperbolicity of the system is “artificial,” a numerical treatment based on the “artificial” eigensys- tem will depend on the choice of artificial compress- ibility. It will be shown that upwind schemes con- taminate the momentum equations in an unbounded fashion through the relation between upwind differ- ence and artificial dissipation. This potentially limits the use of upwind schemes, although in reality this limitation is seldom reached. However, caution must be exercised in the choice of artificial compressibility.

2. Governing Equations For convenience, only 2D inviscid equations will be

analyzed in this paper. The viscous terms are a sim- plified set of their Compressible counterparts and are not affected by artificial compressibility. The conclu- sions of the 2D analysis are directly applicable to 3D equations. All variables in the following are normal- ized by some reference values:’ the freestream velocity, density, viscosity and unit length. The pressure is nor-

The incompressible Enler equations with artificial compressibility are written in the Cartesian z,y and time t system as

malized as (P- ~ ~ ~ j ) / ( p ~ ~ j $ ~ , ) .

(1) wind (including TVD) schemes embedded in a multi- a - 8 - a - ,Q + -E+ --F = 0. aZ ay zone, structured-grid book-keeping framework. It per- mits both steady-state and time-accurate computa- tions using either time-marching, space-marching or space-relasatinn solution techniques. It includes AD1 and various Gauss-Seidel relaxation methods using ei-

mulation for the implicit side operator. The algorithm

1 Member of ‘Technical Staff; currently Assistant Prof., Institute of Aeronautics and Astronautics, National Cheng-Kung Univ., Tainan, Taiwan, ROC. Membet A I A A .

1

where

- (;) , E = ( / i P ) , F = ( :: ) ther the full-block formulation or the diagonalized for- Q = v 2 + p -

~ ~ ~ ! ~ ~ ~ h s g ~ r n c c i i ~ n IIISIIIUIC of AcionauliCI and A ~ ~ ~ ~ ~ ~ ~ ~ ~ c ~ , i n i . . 1989. 411 rishii rcrcwcd. hlanager, CFD Dept., Member AIAA.

In the above, p9 u and u are pressure and velocity com- ponents, and 0 is the artificial compresibility which

nonsingular formulation which requires metria infor- mation only in the considered direction is given in the

couples the continuity equation and the momentum Appendix. . ' equations together to form a hyperbolic equation set. The divergence-free requirement is met when the artifi- cial time term of pressure in the continuity equation is driven t o zero. Conventionally, Eq. (1) is transformed into the generalized coordinates [, v and T as

(2) a a a

ar a( av -Q+ -E+ -F = 0.

where

w 3. Finite-Volume Implementation

The detailed description of the computational algc- rithm can be found in reference papers 7 and 8. Here only those pertinent to the discussion will be men- tioned. Basically i t is a finitevolume implementation of MUSCL type upwind flux-difference schemes. The starting point is the semidiscrete form of Eq. (2):

Q = j Q where j, it are the indices for volume vertices in ( and v.

E t - E=- E v - E = -jQ+ -E + -F J J The numerical flux at a cell surface m + 4 in direction m is given by

F = !kQ+ EE + !!!.F - $1 + - J J J j m + + E ;i[f('Irn++)+f(q~++)-IAI('I,t:-q,+ 1 .

(4) J is the transformation Jacobian and E=, etc. are the transformation metria. where q:t+ and p i t i are the reconstructed right and

Equation (2) has been used to compute both steady- state and unsteady solutions. However, a careful ex- left states of cell surface m + $, and is the flux amination' shows that Eq. (2) does not permit divergence- Jacobian of some intermediate state T which satisfies free solution in a moving grid system. With nonzero & the following Roe's properties: and qt in the continuity equation; the solution would be unphysical even in the steady-state. To recover the ( i ) x = A if q + = q - = F = Q divergencefree field, the flux vectors need to be mod- ified by a diagonal matrix I d = diag(O,l , l) as (4 X(9+ - 'I- ) = f (q+ ) - f('I- ) v

For incompressible variables, it can be proven that the only state T which satisfies these properties is the sim-

For a first-order upwind scheme, & + = Qrn+l and

E = $ I d G + &E+ h7 v=- v u -

J J J ple average of qt and 9 - .

J J J F = I I ~ Q + - E + - F rl

This is equivalent to applying the artificial compress- ibility concept the grid transformation is done instead of before. This is natural since the Cartesian system is only a subset of the generalized system. The latter is closer to the integral form of the conserva- tion law. This modification is of vital importance for problems with moving components such as a turbine cascade.

The modified flux Jacobians and their eigensystems are given in the Appendix. It should be mentioned here that while the choice for the 2D eigenvectors is very straightforward, the choice for the 3D eigenvec- tors has more freedom and hence requires more gener- ality. As noted by Rogers and K ~ a k , ~ the formulation given by Hartwich and Hsu' can become singular for certain values of metria. Rogers, Chang and Kwak'O utilized the reciprocal base vectors in their formulation which requires metr ia information in all three direc- tions to be known a t all times. This places some unde- sirable and unnecessary constraints on both computer time and memory. In this study, a simple, general and

= Qm. For higher order schemes, the right and

left states are reconstructed using information from more than one neighboring cell. The reconstruction process can be limited to obtain TVD properties if so desired. For incompresible flows, this TVD limiting is unnecessary and sometimes undesirable since it de- grades the solution accuracy at local extremas.

4. Time Discretization

written as An implicit time discretization of Eq. (3) can be

Q"t1- Q" + Ar el(Ej++,k- Ej-i,kY'+' + ( F j , k + + - Fj ,k -+Y ' ' ' I+

(1 - o)[(Ej++,k - E j - + , k ) " + ( F j , k + + - F j , k - + ) " I = 0.

where n is the time index, AT is the time step and 8 determines the accuracy of the time discretization. - I t is first-order explicit for 8 = 0, first-order implicit for 8 1 and second-order for 8 = 0.5. By applying

(5)

' Ref. 9 has obtained the same conclusion independently.

2

Newton's procedure and linearizing Eq. (5) about a known state Q', the implicit iterative procedure can be written as

I a a AT aQ aQ { - + O[-(AjE') + -(AtF")])(Q'+' - Q') =

I AT

-{-(Q" -Q")+OIAjEb+AkFa]

with A j E I Ej++,t - Ej-$,k

A t F Fj,t++ - Fj,t-+

Here I is the identity matrix and s is a subiteration index. For the comprgsible flow algorithm, Eq. (6) is valid for both timeaccurate and steady-state compu- tations. It can be sub-iterated to obtain a better a p proximation Q a t ' of Q"+'. However, for incompress- ible flows Eq. (6) does not guarantee a divergence-free solution unless the iteration in n converges.

Since the continuity equation does not need time- accuracy, it can be treated independently by the first- order implicit scheme which has a better convergence property. The divergencefree requirement a t every time lever n + 1 can be obtained by

+ ( I - Is)[AjE" + AtF"]) (7)

with

I, = diag(--, 1, l) , Id = diag(0,1, l) , I , = diag(l,O,O)

Here w is a relaxation parameter to accelerate the con- vergence of the continuity equation. Usually i t should be greater than one. Equation (7) is valid for both time-accurate and steady-state computations. S u b iteration on s can be done to obtain a divergencefree solution at every new time level n + 1. Equation (7) is to be inverted by various space-marcbing or space relaxation methods. Among the existing capabilities of the present code the choices are the AD1 method and various Gauss-Seidel methods in either full-block or diagonalized formulation.

5. Artificial Compressibil i ty a n d Artificial Dis- sipation

Initially, the amount of artificial compressibility p seems to be arbitrary and can be choscn at will. In fact, it is oftten stated that without considering the convergence rate, a large 0 should be chosen so that

1 w

3

the "true" incompressible flow equations can be re- covered. In practice, @ is usually set between 1 to 10 to avoid large factorization errors in multidimensional problems.2*6~'0

However, i t should be pointed out that p can be arbitrary only in the differential equation level, or in the difference equation level when central difference is used. In these cases, @ appears only in the continuity equation, not in the momentum equations. The mrr mentum equations see the effects of @ only through the time variation of pressure. Once the pressure con- verges, the continuity equation and the momentum equations are completely decoupled and the value of p has no effect on the final solution. But this is not true for an upwind scheme. In upwind schemes, the upwind difference is directly related to the eigensystem of the problem and is strongly affected by the choice of @. Even in the steady-state solution, the continuity and the momentum equations are still coupled by @ and hence, its choice will affect the final solution.

To show this, it is necessary to consider the nota- tion that an upwind difference is equivalent to a central difference plus a dissipation. In the definition of nu- merical flux in Eq. (4), this dissipation is represented by the term IAI(qc - q - ) = IA/Aq = RIAILAq. With some algebraic manipulation, it is not difficult to obtain the following expressions in the Cartesian system for z and y directions:

\ { fA"(p + $) + cAu) When four cell surfaces j+ $,j- $ , k + 4 and t - f are considered, each uA'' in the above vectors amounts t o a dissipation operator. I t is a first-order second- derivative dissipation in a first-order upwind scheme, and a third-order fourth-derivative dissipation in a second- order upwind scheme. It can be Seen that @ has con- taminated the momentum equations through the arti- ficial "speed of sound" c. Furthermore, as p increases, the following terms will grow without bound like fl:

1 ! [ A r z ( ~ + $ ) + A y Y ( p + $ ) 1 -! 2 ( cAzzu

CAYYV

where Azz and Ayu represent dissipation operators. The first element is harmless because p can be fac- tored out with the divergence part of the equation. The remaining two elements in the momentum equa- tions represent a poteiitial limitation on the upwind

approach. Unless Azzu and Ayuu both are zero, which is generally not true, these terms will dominate the SD lution for very large 8. Fortunately, in reality A..u and Ayyu are seldom large. They are even smaller for high order upwind schemes. It is for this reason that accurate solutions can still be expected with a reason- able choice of p. 6. Characteristic Boundary Conditions

One of the difficulties in solving the incompressible flows is that generally no Dirichlet boundary condition for pressure exist. The pressure terms in the momen- tum equations exist in the form of pressure gradient, and Neumann type boundary conditions are generally needed. This difficulty can be alleviated by using the characteristic boundary treatment where "extrapola- tion" is used in a natural manner. Taking the far-field boundary condition as an example, characteristic vari- ables can be either extrapolated from flow interior or from the far-field values depending on the sign of the eigenvalues. For example, an inflow in the positive di- rection has two positive going characteristics and one negative going characteristic. This can be written in terms of the left eigenvector L as

(1:; ::: ") (i) - - 132 133 boundary

1 (Illp + 112u + ~13~)freestresm

( 1 2 1 ~ + 1 2 2 ~ + /23u)intc+x ( ((31P + b2'J + h3u)hterior

Similar equations can be formed for the outflow bound- aries with two characteristics extrapolated from i n t e rior and one from the far-field values. These treat- ments provide excellent non-reflective far-field bound- ary conditions. They were used for most of the vali- dation runs presented in this paper.

For an inviscid wall boundary, the velocity tangen- tial condition can he used to replace one of the three characteristic equations while the remaining two are treated by extrapolation from flow interior. It can be written as

;; k ) (F) =

1 ( -(7lt)."all

wall

( I l lP + /IZU + l13u)mterror

( / Z I P + 122U + b23V)mfcnor

This was used for the inviscid flow over a circular cylin- der and a 3D sphere.

7. Validation Results Flat-Plate Flow

The boundary conditions used for flat-plate flow are n-slip and zer-pressure gradient conditions on the plate, characteristic boundary conditions for inflow z and far-stream y boundaries and extrapolations for outflow z boundary. The laminar results for Re = 10,000 were compared with the self-similar Blasius s- lutions in Fig. 1. The computed results are taken from an z station far downstream of the leading edge. The comparison shows excellent agreement.

The algebraic Baldwin-Lomax turbulence model was used in the turbulent computation. Figures 2 and 3 show comparisons of the computed boundary layer profiles and skin friction with experimental data." They show good agreement in general, although some important differences can be clearly observed. In par- ticular, the computed skin friction near the leading edge (z = 0) region is too low. Among the possible causes for this mismatch is the large Azzu term in this region which in turn will cause large dissipation error as discussed previously. 8 used in this computation is 10. For both laminar and turbulent computations, the computed pressure field shows very little pressure variation even around the highly grid-clustered leading edge.

It should he mentioned here that this flat-plate flow problem and some other test problems were also com- puted using a moving grid system. The grid velocity was set arbitraily in three directions. Identical solu- tions were obtained as the stationary grid cases, but the initial convergence was affected due to the differ- v ences in the initial conditions. Inviscid Cylinder Flow

d

A 2-zone grid was used for flows over a circular cylin- der, as shown in Fig. 4. The grid x'as used in both viscous and inviscid computations. The second zone was used mainly to capture the viscous vortex wake better. It is not necessary for the inviscid computa- tion. The far-field was 10 diameters away in zone 1 and extended to 20 diameters in zone 2. Inflow and outflow characteristic boundary conditions were c h o sen automatically based on the sign of local normal velocity. The inviscid wall boundary condition men- tioned previously was used. Figure 5 shows the com- parison of pressure coefficient with ideal flow solution for 8 = 10. Figure 6 shows the results for p = 100. The effect of on the accuracy of local solution around the downstream stagnation point is obvious. The far-field boundary point values (the horizontal line) show very little dependence on 0. Figures 7 and 8 are the com- puted pressure contours. Besides the clear differences caused by 8, the contour lines are very symmetric as they should he.

Oscillating Flat-Plate Flow

The first unsteady calculation done on an infi- nite flat-plate oscillating in I direction with a velocity -

u p = uocos(nt)

The analytical velocity profile of the viscous flow on top of the plate is a damped harmonic oscillation:

u(y,l) = uoe-"cos(nf - 9 )

where 9 = y a , Y is the kenematic viscosity and n = 1 is used in this test.

Only one cell in z direction is needed for the calcu- lation since there is no gradient in z. The z boundary values were extrapolated from interior instead of us- ing the exact solution, although the latter provides the boundary condition in y and the initial condition. The first-order timeaccurate solutions at different times for one half cycle are compared with exact solutions in Fig. 9. Good agreement was obtained in general.

Depending on the time step, the sub-iteration in s needs to be adjusted to reach the desired convergence. The current time step is chosen 0.001 while the period of the oscillation is 2a. For such a small time step, s = 2 is sufficient to maintain a divergence free field. Here the time is normalized by up and unit length.

Viscous Vortex Shedding Behind Circular Cylinder

The viscous vortex shedding behind a circular cylin- der was also computed. Figure 10 shows the overall flow field and the oscillating wake captured by the 2-zone grid. The Reynolds number based on the di- ameter and freestream velocity is 2000. p was set to 10; the sub-iteration s was 3. Only first-order time- accurate solutions were computed. The vortex shed- ding was self-triggered after a large number of iteration in the steady-state where the solution is symmetric. The steady periodic motion of pressure at 8 = 214.5' is recorded in Fig. 11. Here 0 is measured from the upstream nose in clockwise direction. A Fast Fourier Transformation (FFT) was done to capture the domi- nant frequency of the period motion. Figure 12 is the FFT result which shows a major peak at a Strouhal number (ST) 0.2346i 0.0117. The a ~ c e p t e d ' ~ experi- mental data is 0.21. Figures 13- 17 are streamlines in sequence for one complete cycle. The nondimensional time T is normalized by the freestream velocity and the cylinder diameter.

3D Inviscid Flow Over a Sphere

The axisymmetric inviscid flow over a 3D sphere was chosen as the first 3D test. The grid was constructed by rotating the 2-zone cylinder grid about its 0 = 0' and 180' axes in the third direction (4). Since the flow is axisymmetric, only 0" 5 8 5 180° is needed and one cell in 4 (1") is sufficient. The inviscid wall and outer boundary conditions are similar to the 2D cylinder case. The velocity components normal the 4 boundaries were set to zero to simulate the axisym- metric condition. No-flux condition was used for the

Figure 18 shows the surface pressure coefficient com- pared with the analytical solution. The agreement is generally good except near the downstream stagnation point, just as in the 2D cylinder case. I t is expected that a better agreement may be obtained by reduc- ing the amount of p. Figure 19 shows the pressure contours. 3D Square Channel Flow

A fully developed channel flow was computed and compared with the analytical solution. The pressure gradient in z direction was 5 and the Reynolds number was 1 based on channel width and unit velocity. The grid has only one cell in z and 30 cells in the other two directions. Since the flow was fully developed, the v e locity has no gradient in z. No-slip and zerc-pressure gradient boundary conditions were applied to channel walls. Figure 20 is the streamwise velocity profile along the centerline. I t shows excellent comparison with the analytical solution. Figure 21 shows the computed ve- locity contours which are perfectly symmetrical as they should be. 8. Conclusions

A unified computational method originally devel- oped for compressible flows has been implemented to solve the artificial compressibility equations for 2-D and 3-D incompressible flows. The versatile capabil- ities of the compressible flow algorithm are readily available for incompressible flows. These include the finitevolume implementation of high accuracy upwind schemes, multi-zone structured-grid book-keeping frame- work, unified formulation for steady-state and time- accurate solutions using various relaxation techniques and so on. The continuity equation was modified to allow general grid motions. A simple, general and non- singular eigensystem for 3D flux Jacobians was derived which requires metrics information in the considered direction only. The iterative process based on New- ton's procedure was modified to satisfy the divergence- free requirement in a time-accurate fashion. Several steady-state and unsteady computations for 2D and 3D flows were presented for validation.

The relationship between artificial compressibility and upwind difference was discussed. Contrary to pop- ular belief, artificial Compressibility has contaminated the momentum equations through the artificial dis- sipation inherented in upwind schemes. Fortunately, the contamination is not serious for smooth-varying solutions and is even smaller for high order upwind schemes. Although this represents a potential limita- tion for upwind schemes, accurate solutions can still be expected for a reasonable choice of artificial com- pressibility.

Acknowledgements 0 = On and 180" axes. That is, pressure and axial velocity were extrapolated and two non-axial veloci- ties were set to zero. p was chosen 5 b a e d on the 2D cylinder experience.

This work is supported by the Int,ernal R&D fund from Rockwell International Science Center. The au- thors wish to thank every member of CFD group at Science Center for many fruitful discussions.

.-

5

References (1) Chorin, A. J., “A Numerical Method for Solv- ing Incompressible Viscous Flow Problems,” Journal of Computational Physics, Vol. 2, 1967, pp. 12-26.

(2) Steger,J. L. and Kutler P., “Implicit Finite-Difference Procedures for the Computation of Vortex Wakes,” AIAA Journal, Vol. 15, No. 4, April, 1977, pp. 581- 590. (3) Kwak, D., Chang, J. L. C., Shanks, S. P., and Chakravarthy, S. R., “An Incompressible Navier-Stokes Flow Solver in Three-Dimensional Curvilinear Coordi- nate System Using Primitive Variables,” A I A A Jour- nal, Vol. 24, No. 3, March 1986, pp. 390-396.

(4) Chakravarthy, S. R., “Numerical Simulationof Lam- inar Incompressible Flow within Liquid Filled Shells,” Report ARBRGCR00941, U.S. Army Ballistics Re- search Laboratory, Aberdeen Proving Ground, MD, Nov. 1982. (5) Merkle, C. L. and Athavale, M., “Time-Accurate Unsteady Incompressible Flow Algorithms Based on Artificial Compressibility,” AIAA paper 87-1137, CFD Conference, July 1987. (6) Hartwich, P. M. and Hsu, C. H., “High Resolu- tion Upwind Schemes for the Three-Dimensional In- compressible Navier-Stokes Equations,” AIAA paper 87-0547, Reno, January 1987.

(7) Chakravarthy, S . R., “The Versatility and Relia- bility of Euler Solvers Based on High Accuracy TVD Formulations,” AIAA paper 86-0243, Reno, January 1986. (8) Chakravarthy, S. R., Szema, K. Y., and Haney, J . W., “Unified “Nose-To-Tail” Computational Method for Hypersonic Vehicle Applications,” AIAA paper 88- 2564, Applied Aerodynamics Conference, Williamsburg, Virginia, June 1988.

(9) Rogers, S. E. and Dwak, D., “An Upwind Dif- ferencing Scheme for the Time-Accurate Incompress ible Navier-Stokes Equations,” AIAA 88-2583-CP, 6th Applied Aerodynamics conference, Williamsburg, Vir- ginia, June 1988.

(10) Rogers, S. E., Chang, J. L. C. and Dwak, D., “A Diagonal Algorithm for the Method of Pseudocom- pressibility,” Journal of Computaiional Physics, Vol. 73, 1987, pp. 365-379. (11) Choi, D. and Merkle, C., “Application of Time- Iterative Schemes to Incompressible Flow,” AIAA Jour- nal, Vol. 23, No. 10, Octorber 1985, pp. 1518-1524.

(12) Weighardt, K., “Flat Plat Flow,” Proceedings of the AFOSR-IFP-Stanford Conference on Computation of Turbulent Boundary Layers, D. E. Coles and E. A. Hirst , Editors, Vol. 2, 1968, pp. 98-123.

(13) Blevens, R. D., Flow-Induced Vibration, 1986.

A P P E N D I X I. E I G E N S Y S T E M OF 2D FLUX J A C O B I A N

I/ 0 Pk, OkY kz uk,+U uk, ) k, vkz v k , + U

A- 0 0 A = ( : An 0 A+ 0 )

where

h=C,11

(I = kt + uitz -b vk,

e = Jm 1

* - 2

A0 = u

u’ - -k, + uitz + vky

A- = u; - c

A+ = u; + c

For the right (R) and left ( L ) eigenvectors, all metrics

are normalized by

a+c+ o - A - C u-A+k, k, u-A-k= v - A c k , -k, v - A - k ,

1 2

c- = c - -k, 1 2

e+ = c + -it,

11. E I G E N S Y S T E M O F 3D FLUX J A C O B I A N

0 Ok, PkY

I ; , V L vk, + u l ik,

I ; , u k , i U uk,

I ; z wk, wky wk, i I/

A = ( 6

A - 0 0 0

A = ( ! ; XO. :) 0 0 A+

where

k = t,nC U = k, +uk,+uky+ wk,

1 2

u:, = -k, + ut, + vk, + wkz

A - = u : , - c

A+ = u:, + c

A0 = u

For the right ( R ) and left ( L ) eigenvectors, all metrics

are normalized by

1 2

c - = c - - k ,

1 2

C+ = c + -k,

where Qj 's are defined by two nontrivial vectors g2 and d3 satisfying

Here we suggest

The elements of the left ( L ) eigenvector are:

7

&"e.oan

Fig. 1 Laminar flatplate boundary layer profile compared with Blasius solution (Re = 10.000).

w*,

" "1 I

Fig. 2 compared with Weighardt's data (Re = 2.2 x 10').

Turbulent flatplate boundary layer profile

u m .

o.l 1

0.8 I - C O M W E C 0 WEIGHARDT

0.0 1 0.0 1 .o 2.0 3.0 4.0 5.0

x IrnJ

Fig. 3 Turbulent flat-plate skin-friction C, corn. pared with Weighardt's data (Re = 2.2 x lo6).

-15.01 1

-10.0 -4.0 2.0 8.0 14.0 20.0 X

Fig. 4 2-zone grid for circular cylinder. .e.._

2.0

DOWNSTREAM

1.0

-1.0

CP

-2.0

-3.0

-4.0 I I 0.0 80 120 180 240 300 360

9

Fig. 5 Surface pressure coefficient C, for inviscid flow over a cylinder (p = 10).

r.. l l . 2.0r 7 I

v

-4.0 0.0 80 120 180 240 300 360

e

Fig. 6 Surface pressure coefficient C, for inviscid Row over a cylinder (0 = 100). v

1 . 2

0.4

Y

4 . 4

-1.2

-2.0 -2.0 -1.2 4,. 0.4 1.2 2.0

x

Fig. 7 Pressure contours for inviscid flow over a cylinder (0 = 10).

2 .o

1.2

0.4

" 4..

-1.2

-2.0

u**

\ /

\

0 -1.2 4 . 4 0.4 1.2 2.0 x

Fig. 8 Pressure contours for inviscid flow over a cylinder (0 = 100).

Y., 10.0

8.0 - caMwrEa SYMBOLS: EXACT

6.0

Y

4.0

2.0

0.0 -1.0 -0.6 -0.2 0.2 0.6 1 .o

u

Fig. 9 Velocity profiles at different times for Row Y over an oscillating plate.

U".. . - 70 30

15 01

V

-15.0 -10.0 -4.a 2.0 8.0 14.0 20.0

X

Fig. 10 cylinder ( R e d = 2000).

Viscous vortex shedding behind a circular

a - 214 .5

-0.6

-3.0 -2'41 50.0 55.0 60.0 65.0 70.0 75.0

Fig. 2000),

.i ? I Y P

- E - 5

11 Time history of C, (0 = 214.5',&,, =

STRWHAL NUMBER

Fig. 12 FFT result for viscous vortex shedding behind a cylinder (Re,, = 2000).

-1.81

1.8-

-3.0 -1.0 0.2 1.4 2.6 3.8 5.0

X

Fig. 1 3 Streamlines for viscous vortex shedding behind a cylinder (T = 7 5 . 5 4 . R ~ = 2000).

UY

1 - 76.55 I - 79.58

-1.81

UPSTREAM W W N I T R L I I M ST*C1(I\"ON STl \GNlnaN

-3.01 -1.0 0.2 1.4 2.6 3.8 5.0

X

Fig. 14 Streamlines for viscous vortex shedding behind a cylinder ( r = 76.55,Red = 2000).

U I I

T - 7'1.56

1.8

-3.0 -1.0 0.2 1.4 2.6 3.8 5.0

Fig. 15 Streamlines for viscous vortex shedding behind a cylinder (T = 77.56,Red = 2000.).

X

-1.5 I 0 36 72 108 tM 380

8

Fig. 18 flow over a 3D sphere (0 = 5).

Surface pressure coefficient C, for inviscid

10

a 0

3 2

1 4

1

1.6

0.8

0.C 2.0 -1.2 *.4 0.. 1.2 2.0

x

Fig. 19 Pressure contours for inviscid flow over a 3D sphere ( f l = 5).

U". 0.5

(1.3 rw\ *.'V, TiMcT . COMRmD

-0.5 0.0 0.1 0.2 0.3 0.4 0.5

U - V E L O r n

Fig. 20 viscous square channel flow (Re = l.,P= = -5.).

Centerline velocity profile for well-developeu

Fig. 21 Velocity contours for well-developed vis- cous square channel flow (Re = l.,Pr = -5.).

11