Mitsuhiro Murayama and Kazuhiro Nakahashi- Numerical Simulation of Vortical Flows Using Vorticity Confinement Coupled with Unstructured Grid

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NUMERICAL SIMULATION OF VORTICAL

FLOWS USING VORTICITY CONFINEMENT

COUPLED WITH UNSTRUCTURED GRID

AIAA-2001-0606

Mitsuhiro Murayama and Kazuhiro Nakahashi

Department of Aeronautics and Space Engineering

Tohoku University, Sendai, JAPAN

Shigeru Obayashi

Institute of Fluid Science

Tohoku University, Sendai, JAPAN



American Institute of Aeronautics and Astronautics 1

AIAA-2001-0606

NUMERICAL SIMULATION OF VORTICAL FLOWS USING

VORTICITY CONFINEMENT COUPLED WITH UNSTRUCTURED GRID

Mitsuhiro Murayama*, Kazuhiro Nakahashi

†

and Shigeru Obayashi

‡

Tohoku University, Sendai 980-8579 Aoba-yama 01, JAPAN

ABSTRACT

This paper discusses the use of the vorticity

confinement method coupled with the unstructuredgrid approach to simulate vortical flows. The method

is evaluated by several vortical flow computations of

the leading-edge separation vortices on delta wings

and the wing tip vortices of NACA0012 wing. It is

shown that the vorticity confinement can keep the

vorticity away from the numerical diffusion

effectively. Although further study to reduce the

dependency of the confinement coefficient on thegrid density is required, the present results indicate

the possibility of accurate vortical flow computations

by the vorticity confinement method coupled with

unstructured and adaptive refinement grids.

1. INTRODUCTION

With the recent progress, the computational fluiddynamics (CFD) is very close to its matured stage in

the computation of flows around airplanes at

designed conditions. However, it is still difficult to

deal with complex flows at off-design conditions

where flow separates and vortices characterize theflow features. These vortical flows are often

encountered at various important engineering

problems. For example, flows around a delta wing at

high incidence are characterized by the leading-edgeseparation. This separation vortex generates a large

non-linear lift increment called vortex-induced lift at

moderate angles of attack. At higher angles of attack,

however, this vortex is to burst, resulting in a suddendecrease of the lift. The BVI (Blade-Vortex

Interaction) problems and vibratory loading problems

are also caused by impingements of vortices on

helicopter rotor and aircraft fuselage. Wing-tipvortices of airplanes during the take-off and landing

are another serious problem to deal with the

congestion at airports accompanied with the rapid

growth of the aviation.

* Graduate student† Professor, Department of aeronautics and space engineering,Associate Fellow AIAA‡ Associate professor, Institute of Fluid Science, Senior Member Copyright © 2001 by American Institute of Aeronautics andAstronautics, Inc. All rights reserved.

Accuracy of the vortical flow simulations,

however, is still not good enough. Generally, the

computational grid becomes rapidly coarser as it

becomes far away from the body surface. There, the

vortices are highly diffused due to the numericaldiscretization error. By the use of a highly dense grid,

numerical dissipation of the vortices may be

minimized. However, flow computations around 3-D

complex bodies with large-scale separations andvortices are difficult within the realistic number of

grid points.

One approach to crisply capture the vortex is to

use the adaptive grid method. We proposed an

effective and efficient adaptive grid refinement

method for the improvement of grid resolution

around a vortex center using the vortex center identification method. The resulting method was

applied to flows around a delta wing and showed

significant improvements in resolution of the

leading-edge separation vortices [1]. For further improvements, however, not only the way to reduce

dissipation by the refinement of the grid but also the

model to keep the vortices from diffusion will be

needed.

“Vorticity confinement method” has been

proposed to reduce the diffusive property of the

vortical flow simulations [2-4]. In the method, thesource term added to the Navier-Stokes equations

works as it convects the discretization error back into

the vortex center and thus confine the vortex. The

method is applied to flows around helicopter rotors

with fuselage, and shows reasonable improvements

in the vortex resolution [3]. However, the method is

still needed to be tuned for the confinement

parameter at a particular grid. In addition, the

vorticity confinement has been used on Cartesian

grids or structured surface conforming grids, not on

unstructured grids. Unstructured grids will be more

suitable for the numerical simulation around 3-Dhighly complex bodies.

The objective of this paper is to develop the

vorticity confinement method coupled with the

unstructured grid method. The capability of theresulting method is evaluated by the computations of

the following four flow fields with vortices,






whereijS∆ is a segment area of the control volume

boundary associated with the edge connecting points

i and j . The term h is an inviscid numerical flux

vector normal to the control volume boundary, and±

ijQ are values on both sides of the control volume

boundary. The subscript of summation, )(i j , refersto all node points connected to node i .

The Harten-Lax-van Leer-Einfeldt-Wada Riemannsolver (HLLEW) [9] is used for the numerical flux

computations. Second-order spatial accuracy is

realized by a liner reconstruction of the primitive

variables T pwvu ],,,,[ ρ =q inside the control volume,

viz.,

)(),,( iiii z y x rrqqq −⋅∇Ψ+= (6)

where r is a vector pointing to point ),,( z y x ; and

i is the node number. The gradients associated with

the control volume centroids are volume-averagedgradients computed using the value in the

surrounding grid cells. A limiter, Ψ , is used to make

the scheme monotone. Here Venkatakrishnan’slimiter [10] is used because of its superior

convergence properties.

To compute viscous stress and the heat flux terms

in )(QG , spatial derivatives of the primitive

variables at each control volume face are evaluated

directly at the edges.

A one-equation turbulence model by Goldberg-

Ramakrishnan (G-R)[11] was implemented to treatturbulent flows. This model does not involve wall

distance explicitly so that it is of great benefit to

unstructured grid method.

The lower-upper-symmetric Gauss-Seidel

(LU-SGS) implicit method originally developed for

structured grid is applied in order to compute thehigh Reynolds number flows efficiently. The

LU-SGS method on unstructured grid can be derived

by splitting node points )(i j into two groups,

)(i L j∈ , and )(iU j∈ for the first summation in

LHS of Eq. (5). With nnQQQ ∆−∆=∆

+1 , the final

form of the LU-SGS method for the unstructured grid

becomes the following two sweeps:

Forward sweep:

])(5.0[)(

¦∈

∗∗−∗∆−∆∆−=∆

i L j

j A jiji S QhRDQ1 ρ (7a)

Backward sweep:

¦∈

−∗∆−∆∆−∆=∆

)(

)(5.0iU j

A jijii S j1

QhDQQ ρ (7b)

where )()( QhQQhh −∆+=∆ , and

¦ ∆+=∆ )(

)5.0(i j Aijt

V S j ID ρ , (8)

ii

i j

n

ijij

i j

ijijiV SS SGhR +∆+∆−= ¦¦ ∗

)()(

(9)

The term D is diagonalized by the Jameson-Turkel

approximation [12] of the Jacobian as

)(5.0 IAA A ρ ±=± , where A ρ is a spectral radius of

Jacobian A .

The lower/upper splitting of Eq. (7) for the

unstructured grid is done by a grid reordering

technique [13] that was developed to improve theconvergence and the vectorization. For unsteady flow,

the time accuracy of the LU-SGS solution algorithm

is recovered by the Newton iteration using

Crank-Nicolson method.

5. RESULTS

5.1 CASE1: Single Vortex in Freestream

For the validation of the present method, a 2Dsingle vortex in freestream was computed first. The

formulation of the initial vortex suggested in Ref. [4,14] is written as follow,

( )¯®-

≤≤+

<=

0,

,

Rr Rr B Ar

Rr Rr U r u

c

cccθ (10)

220 c

cc

R R

RU A

−

−= (11)

220

20

c

cc

R R

R RU B

−

= (12)

( ) 00 =θ u , ( ) cc U Ru =θ , ( ) 00 = Ruθ (13)

where 0 R is an outer radius and Rc is a core

radius of the vortex. cU is a maximum core velocity.This initial vortex conditions are added to freestreamconditions.

Solutions were obtained at a freestream Mach

number of 5.0=∞ M . The Reynolds number is

1.2×106. The initial vortex conditions were set to

∞=U U c , 05.0=c R , c R R ×=100 . The outer

boundary is a square whose edges have 1.0 length

and periodic boundary conditions are applied. Two

types of the unstructured computational grid wereused as shown in Fig. 2. Grid 1 which has 14,094

node points has approximately uniform cells. Grid 2

which has 18,024 node points is constructed by the

division of the Grid 1 and has relatively dense region

in the middle of the grid and non-uniform grid

density.

The variations of the maximum vorticity

magnitude around vortex center and vorticity

contours with/without the vorticity confinement are

shown in Figs. 3 and 4, respectively. The centered

vortex moves downstream and passes through thedownstream boundary and reappears from the other

upstream side and moves to the center of the grid

again. This process is defined as one cycle in these

computations.




(a) Grid 1 (14,094 nodes)

(b) Grid2 (18,024 nodes)

Fig. 2. Computational grid

0

10

20

30

40

50

60

70

0 10 20 30 40 50

Original ResultEPS0.001EPS0.003EPS0.005EPS0.01

V o r t i c i t y M a g n i t u d e

Cycle (a) Grid 1

0

10

20

30

40

50

60

70

0 5 10 15 20 25 30

Original ResultEPS0.001

EPS0.003

EPS0.005

EPS0.01

V o r t i c i t y M a g n

i t u d e

Cycle

(b) Grid 2

Fig. 3. The variations of the maximum vorticitymagnitude around vortex center with/without the

vorticity confinement (EPS: the value of the vorticityconfinement parameter, ε )

(a) (b) (c)Without confinement

(a) (b) (c)

With confinement (ε=0.003)

Fig. 4. The vorticity contours on Grid 1 with/without

the vorticity confinement, (a) after one cycle, (b)

after ten cycles, (c) after fifteen cycles

In the case without the vorticity confinement, thevortex rapidly diffused. In the case with the vorticity

confinement with moderate values of the

confinement parameter ε, it can be seen that the

degrees of the dissipation of the vortex are decreased

by the confinement method and the strength of the

vorticity is preserved even after 50 cycles.

However, this effect is influenced by the values of

the confinement parameter ε. Too large values of the

parameter may lead to the non-physical results.

Moreover, in the case using the non-uniform grid,Grid 2, the results are fluctuant, especially when therelatively large value of the parameter ε is employed.

These results suggest that the effect of the

confinement highly depends on the grid density and

the confinement parameter.

5.2 CASE2: Vortical Flows Around a Double Delta

Wing

The method was applied to vortical flows on delta

wings with high incidence. The geometry used in the

present study is a double delta wing shown in Fig. 5.The leading-edge sweep angle is 80 degrees at the

strake and 60 degrees at the main wing. The

thickness is 0.6% of the root chord length and the

leading edge is rounded. The first point above the

wing surface is located at a distance of 8.0×10-5

of

the root chord length. The outer boundaries are

located 10-root chord length away from the body

surface. The total number of the initial grid points is676,541. The computational grids in the cross flow

plane at 62.5% chord length are shown in Fig. 6.

Solutions were obtained at a freestream Mach

number of 3.0=∞

M and angle of attack of 12º. The

Reynolds number based on the root chord is 1.3×106.

In this study, laminar flow was assumed.




Fig. 5. Surface grid of the double-delta wing

Fig. 6. Cut view of the grid at x/c=0.625

First, the results without using vorticity

confinement are discussed. Total pressure contours

computed on the initial grid and the adaptive gridrefined twice are shown in Fig. 7. The adaptive grid

has 1,042,292 grid points. In this flow field, two

leading-edge separation vortices originating from the

strake and main wing interact with each other in the

main wing region. The strength of the strake vortex

becomes weaker at the downstream of the kink

because the energy feeding to the vortex from the

strake leading edge will decrease, while the strengthof the main wing vortex increase as the distance from

the kink increases downstream. The weak strake

vortex is moved outward influenced by the velocity

field induced by the relatively strong wing vortex andmerges with the wing vortex eventually. Compared

with the results on the initial grid, the primary and

secondary vortices are more clearly captured by the

adaptive grid refinement method.

For the validation, the surface pressure coefficients

at different axial locations were compared with

experimental data by the Brennenstuhl and Hummel[15] in Fig. 8. Two pressure peaks by the inner strake

vortex and outer wing vortex can be seen in this

experimental data. In the computational results on the

initial grid, however, the inner peak at x/c=0.625 bythe strake vortex is much lower and it can be seen

that the strake vortex has already been weaken by the

numerical diffusion due to the lack of the grid density.

By using the adaptive grid, the diffusion of the strake

vortex is suppressed as shown in Fig. 7. However,

the improvement of the surface pressure prediction is

very small as shown in Fig. 8. The main reason of

this discrepancy may be caused by the fact thatlaminar flow was assumed in the computation

although the laminar/turbulent transition was probably observed in the experiment, as discussed in

Ref. 16.

(a) Initial Grid

(b) Adaptive grid

Fig. 7 The total pressure contours

( 6100.1Re,3.0,0.12 ×=== M α )

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

ExperimentInitial gridAdaptive grid

- C p

Spanwise location (a) Surface pressure coefficients at x/c=0.625

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

ExperimentInitial gridAdaptive grid

- C p

Spanwise location (b) Surface pressure coefficients at x/c=0.75

Fig. 8. Surface pressure coefficients at differential

axial locations without the vorticity confinement( 6100.1Re,3.0,0.12 ×=== M

α )




The results using vorticity confinement are

compared in Fig. 9. The confinement was applied to

the regions without a boundary layer. Computations

were performed with varying the value of the

confinement parameter ε on the initial grid. With

moderate values of the confinement parameter, the

resolution of vorticity is effectively improved.However, the method creates extra correction in the

case that the values of ε are too large.

The total pressure contours with/without vorticity

confinement at x/c=0.625 and 0.75 are shown in Fig.

10. From these results, it can be seen that the vortex

core position is not affected by the confinement and

the vorticity is concentrated to the vortex core

reasonably. The confinement is more effective than

the adaptive grid refinement method even with much

less grid points.

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

ExperimentInitial gridEPS0.005EPS0.01EPS0.05

- C p

Spanwise location (a) Surface pressure coefficients at x/c=0.625

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

ExperimentInitial gridEPS0.005EPS0.01EPS0.05

- C p

Spanwise location

(b) Surface pressure coefficients at x/c=0.75

Fig. 9. Surface pressure coefficients at differentialaxial locations with the vorticity confinement

( 6100.1Re,3.0,0.12 ×=== M α ); EPS is a value of the

confinement parameter ε

(i) x/c=0.625 (ii) x/c=0.75

(a) Initial grid

(i) x/c=0.625 (ii) x/c=0.75

(b) Adaptive grid

(i) x/c=0.625 (ii) x/c=0.75

(c) Initial grid with the vorticity confinement,ε

=0.01

Fig. 10. The total pressure contours with/without

vorticity confinement

5.3 CASE3: Vortical Flows with Vortex

Breakdown Around a Delta Wing

The confinement method was tested for the vortex

breakdown simulations shown in Fig. 11. The

geometry used in this computation is a slender delta

wing of aspect ratio of unity and a sweep angle of 76

degrees. The freestream Mach number is 0.3, angleof attack is 32º, and the Reynolds number based on

the root chord is 6100.1 × . Laminar flow was

assumed. This flow field was numerically simulated

by the present unstructured grid method [1] and the

predicted positions of the vortex breakdown showed

good agreements with experiments of Hummel andSrinivasan [17].

In Fig. 11, the streamlines starting from the wing

apex show the vortex breakdown pattern near the

trailing edge of the delta wing. The results using the

vorticity confinement computed on the same grid and

conditions are shown in Fig. 12. With larger valuesof ε, the vortex breakdown does not appear.

Fig. 11. The computed flow fields around a delta

wing with vortex breakdown

(6

100.1Re,3.0,0.32 ×=== M

α )




(a) ε=0.001 (b) ε=0.005

(c) ε=0.01


wing with vortex breakdown using the vorticity

confinement ( 6100.1Re,3.0,0.32 ×=== M α )

Figures 13 and 14 show the results with large-scale

vortex breakdown at higher angle of attack of 40º. By

the introduction of the vorticity confinement with

large values of ε, the starting points of the vortex

breakdown and its pattern are considerably

influenced. The vorticity confinement gives rotation

components to the vortex and confines the vortex.

However, with larger values of confinement parameter, excessive rotational components are

added and may destroy the flow physics.


wing with vortex breakdown

( 6100.1Re,3.0,0.40 ×=== M α )

a) ε=0.001 (b) ε=0.005

(c) ε=0.01


wing with vortex breakdown using the vorticity

confinement ( 6100.1Re,3.0,0.40 ×=== M α )

5.4 CASE4: Wing Tip Vortex of NACA0012 Wing

Finally, the present method is applied to the

computations of a wing tip vortex of a NACA0012

wing. The geometry used in the present study is a

NACA0012 rectangular wing of aspect ratio 3 as

shown in Fig. 15. The initial grid has nearlyhomogeneous grid density in the wake region and the

total number of the grid points is 701,037. The axial

direction coincides with the wing chord direction.

The outer boundary is a sphere whose radius is

15-root chord length. Solutions were obtained at a

freestream Mach number, M ∞=0.12. The Reynolds

number based on the root chord is 0.9×106

and angle

of attack is 10.0°.

(a) Surface grid (b) Close-up view

(c) Cut view of the fine grid at 95% semi-span

Fig. 15. Computational grid of NACA0012

The vortex centerlines and vorticity contours

obtained on the initial grid is shown in Figs. 16 and

17, respectively. The wing tip vortex far away from

the trailing edge is highly diffused although the

vortex centerlines are identified clearly.

Fig. 16. The vortex centerlines at the initial grid




Fig. 17. Contours of the vorticity magnitude at theinitial grid

Therefore, adaptive grid refinement method using

the vortex center for the concentration of the grid

points around the vortex core to decrease the

diffusion of the vorticity is applied to the initial grid.The grid is refined twice and the resulting number of grid points is 815,262. The results are shown in Figs.

18-20. These results show the improvement about the

vorticity magnitude by the adaptive grid refinement.

More grid refinement may improve the results, but

from the limitation of the computational resources,

the physical model to keep the vortices from

diffusion may be required.

(a) Initial Grid (b) Adapted grid refined twice

Fig. 18. Cut views of the grid at x/c=5.0

Fig. 19. Contours of the vorticity magnitude at the

adaptive grid refined twice

0

2

4

6

8

10

12

14

2 4 6 8 10

Initial GridAdaptive Grid 1Adaptve Grid 2Adaptive Grid 3Adaptive Grid 4

V o r t i c i t y

Distance From Trailing Edge (x/c)

Fig. 20. The variation of the maximum vorticity

magnitude around vortex center

The results using vorticity confinement are shown

in Fig. 21. The confinement was applied to the wakeregions. The vorticity contours with vorticity

confinement on the initial grid are shown in Fig. 22.In the case of the initial grid, the results show some

improvements although the overall level of the

vorticity magnitude is still low due to the poor grid

density of the grid in the wake region near thetrailing edge. However, larger values of the

confinement parameter ε lead to extra effects as

shown in Fig. 22(b).

The results on the adapted grid refined twice with

the vorticity confinement are shown in Figs. 21(b)

and 23. It is apparent that the vorticity was confinedand the method successfully decreased the numerical

diffusion of the vorticity by the coupling of the

adaptive grid refinement method and the vorticity

confinement. By the use of the confinement, the

diffusion of the vortex is suppressed at the stationabout ten chord length away from the trailing edge.

The vorticity contours on a cut view at x/c=5.0 are

shown in Fig. 24. In these figures, it can be seen that

the vortex core position is not affected by theconfinement and the vorticity is concentrated to the

vortex core.

However, the effect highly depends on the value of the confinement constant coefficient, ε and grid

density. Moderate ε improves the results, while the

large ε works extra correction. And in Fig. 21(b),

the computed vorticity magnitude using the vorticityconfinement is fluctuant and not smooth at both

value of ε . This is because the grid size on

unstructured grid is not regular although the adaptive

grid refinement was applied and the grid density

becomes approximately homogenous around vortex

center. The confinement effects may depend on the

grid. From these results, it can be seen that optimal

ε which is different by the place and grid may berequired again.




0

2

4

6

8

10

12

14

2 4 6 8 10

Initial GridEPS0.005EPS0.01

V o

r t i c i t y

Distance From Trailing Edge (x/c)

(a) Initial grid

0

2

4

6

8

10

12

14

2 4 6 8 10

Adaptive Grid 2EPS0.01EPS0.02

V o r t i c i t y

Distance From Trailing Edge (x/c) (b) Adaptive grid refined twice

Fig. 21. The variation of the maximum vorticity

magnitude around vortex center with vorticity

confinement (EPS: the value of the vorticity

confinement parameter ε )

(a) ε =0.01

(b) ε =0.05Fig. 22. Contours of the vorticity magnitude at theinitial grid with vorticity confinement

ε =0.02

Fig. 23. Contours of the vorticity magnitude at theadaptive grid refined twice with vorticity

confinement

(a) Without confinement (b) With confinement

ε =0.02

Fig. 24. Vorticity contours in the cross flow plane at

x/c=5.0 of the adaptive grid refined twice

6. CONCLUSION

The vorticity confinement method coupled with

unstructured grid has been applied to the numerical

simulations of four vortical flows. In the case of a

single vortex in freestream, the effect of the vorticityconfinement on unstructured was validated and

problems about the grid dependency were pointed

out. In the case of vortical flows around delta wings,

the confinement was found more effective than the

adaptive grid refinement method. In the vortex

breakdown case, it was demonstrated that the use of

excessive confinement parameters could suppress the

breakdown and thus destroy the flow physics. Finally,in the case of a wing tip vortex of NACA0012 wing,

it was shown that the confinement method could

suppress the numerical diffusion of the vortex far

away from the trailing edge by the coupling of theadaptive grid refinement method.

The results obtained in this study show that,

although further study to reduce the dependency of

the confinement coefficient, ε on the grid density

is required, the vortex confinement method coupled

with unstructured and adaptive refinement grids has

the possibility of accurate simulations of vortical

flows.




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Documents

Mitsuhiro Murayama and Kazuhiro Nakahashi- Numerical Simulation of Vortical Flows Using Vorticity Confinement Coupled with Unstructured Grid