[American Institute of Aeronautics and Astronautics 9th International Space Planes and Hypersonic Systems and Technologies Conference - Norfolk,VA,U.S.A. (01 November 1999 - 05 November

. -_ (c)l999 American Institute of Aeronautics & AstronauticCTpu~of author(s) and/or author(s)’ sponsoring organization.

: A99-45416 : --------------_ AIAA 99-4838

.- A NUMERICAL TEST OF VISCOUS TRANSONIC FLOWS WITH TVD SCHEME

Xinjun Cui t A.I. Ruban $ Yiyun Wang 0 Maths Department, University of Manchester

Oxfdrd Road,’ Manchester Ml3 9PL, UK 11

Abstract

The two-dimensional compressible unsteady Navier-Stokes equations are computed with a second-order accurate Non-MUSCL TVD scheme to obtain the steady-state solution ac- cording to the’ time-dependent method. The shock wave is successfully captured with a good resolution at both. transonic and supersonic regions, and the capturing process is analyzed. The comparison of the higher-order accurate TVD scheme with the first-order accurate is made in order to demonstrate the difference of the resohrtion to shock waves. The process of approaching the steady state with the time- dependent method is explained. In addition, the interaction of shock waves for different Mach numbers and different angles of attack are described by showing the local flow situation at the interaction region.

Keywords

TVD, transonic, time-dependent, steady-state solution, shock wave

1 Introduction

Because of the complexity of the flow, the compressible Navier-Stokes are a mixed set of elliptic-parabolic

transonic equations equations

for steady flow and a set of hyperbolic-parabolic equations for unsteady flow. For the recent numerical computations of ‘transonic flow, it is therefore much more popular to adopt the time- dependent method to obtain the steady state solution by computing the unsteady compressible Navier-Stokes equations.

The explicit finite difference scheme with second order accurate both in space and time was generated by MacCormark in 1969 and further

developed during 70s. In 1981, MacCormack cre- ated an implicit two-stage method with wider application. Meanwhile, a finite difference implicit method with AD1 (alternating direction implicit) scheme was introduced by Beam and Warming in 1978 which is more efficient for mod- erate to high Reynolds number. For those types of second or higher order accurate schemes, however, there may occur dilliculties such as pseudo- fluctuations near contact discontinuity, nonlin- ear instabilities or choosing a non-physical solution. Moreover, an artificial viscous term gener- ally needs to be added into the schemes in order to get a good convergence. Unfortunately, this process will bring about an irretrievable loss of useful information, thus degrading the resolution to the shock wave.

From the beginning of 80’s with the concept of TVD (Total Variation Diminishing) introduced, a different type of finite difference scheme, known as TVD scheme, was rapidly developed and im- proved. There are two major types:

l Non-MUSCL (Non Monotonic Upstream Scheme for Conservation Laws) developed by Harten and Yee.

,. MUSCL developed by Osher and. Chakravarthy.

Here the second order accurate Harten-Yee TVD scheme is used. For the scheme, because its numerical viscosity is related with the eigenvalues of the numerical flux Jacobian matrix, it varies with the eigenvalues then automatically adjusts the size of the numerical viscosity, thus not only improving the resolution of the scheme to the shock wave but making the scheme more stable and eliminating the pseudo-fluctuations of contact discontinuous region.

2 Method *Visiting Research Fellow tProf. and Chief, Center for Large-Scale CFD of

Manchester University SProf. of Beijing Aerodynamics Research Institute scopyright-01999 by the American Institute of Aero-

nautics and Astronautics, Inc. All rights reserved.

Governing Equations

The two-dimensional compressible Navier- Stokes equations with non-dimensionalization in body-fitted C.S. can be described as follows:

(c)l999 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

Where

0 = Pu (2.2)

B = Jy($zE -i- J&F) (2.3)

fi = J-l(qzE + r,+,F) (2.4) ‘. u = b,mw4T (2.5)

E = b, pu2 + P, ~~21, (e + PW (2.6) F = b, PUV, p2 + P, (e + ~b-4~ (2.7)

J = hly - &?2 = (“EYq - G?Y$l (2.8)

Finite Difference Scheme

For the equation (2.1), its corresponding TVD scheme can be, expressed in the following form. (with At = A7 = 1 and 0 3 8 5 1)

:n+l Afijfk + flAr[(Ej+l/,,k - zJTf/?,k) + 1

:n+l y-n+1 (Fj,lc+lp - Q42)l = RHS (2.9)

Where

RHS =~ Cl- B)AT[(~;+~,~,~ - i6;vl,2,k) + (:;k+1,2 - qin,k-l,2)l + VI-S (2.10)

and Ai?:, = oTk+r - ojnk,. VIS refers to the viscous part of equation (2.i). (multiplied by AT) which is computed with a compact three-point central stencil.

For the terms with similar forms to Z’j+r/s,k, they are:

A Rsj+l/2,k * @j+1/2,k)-(2.11)

~@j+lP,k = [44+112,L). (S~+l,k +gi) - w+1,2,1c + Yj+1,2,k)~* 4+1,2,k ] (2.12)’

where 1 is the number of the flow components. I Xjfl,2,k. terms are the eigenvalues of the matrix

a$ ^* $,k and $+1 k are limiters which are the l?zy parameters. to &<rove the first-order accurate: scheme into second order accurate. Here .Harten’s limiter is adopted as follows,

&k = minmod(a;+,,2,k, &,2,k) (2.14)

with _’ ._

1 S;+l.k-Sj.k -fj+l/2,k = I.’

aj+1/2.~ eif $i+l/P,k # ’ (2.15) 0. if c$+,,; k = 0

“_ .’

(2.16)

For the two-dimensional flow computation, this scheme is finally rearranged into a five-point difference scheme. Then, by ‘making an oper- ator splitting with AF (approximate factoriza; tion), we can compute a block-tridiagonal matrix equations with Thomas technique along < and 77 directions individually. -

3 Necessary- Conditions

Initial and Boundarv conditions

For the initial conditions, the whole field is given with free stream conditions except for the boundary layer. For a.better convergence, the velocity distribution is decided with l/7 exponent distribution law in the boundary, layer.

For the boundary condition, the general viscous conditions on wall surface are given as: u = w= 0 (no-slip condition), and pressure ‘and tem- perature are obtained from. $ = 0, g -= 0 . And the outer boundary and the front boundary conditions. are the same as the free stream conditions. For the back boundary, the corresponding u, II, p and p are determined with parabolic extrapolations from inner points or with charac- teristics method. For the wake. cut, its parameters are averaged from corresponding first- inner points.

Grid Generation ’

The grid is generated into a C - I? type with a mixed numerical and algebraic ,method for

2


NACA 0012. The length of the outer boundary is five times of the body length. The final grid is shown in Fig. 1 and 2. For the computation here, the grid is given with 189 x 61 for Reynolds number Re = 1E5.

6, I

Figure 1:’ C - H type grid for NACA 0012

A,) . f ,.,.,,.......... f . . . . f . . j .._.......... i,.. ..,.......

Figure 2: Enlargement of the grid

4 Computations

Canturing of Shock Waves for Different AL,

With the difference scheme suggested in the paper, the computations are carried out for Mach numbers M, = 0.75,0.82,0.85,0.90,0.95,1.05 and 1.50,2.50 for supersonic flows by giving Reynolds number Re = 1E5, with angle of attack Q! = 0” firstly, and their Mach number contours are shown from Fig. 3 to Fig. 10.

From the figures, at transonic region, a normal shock wave is captured at iV&, = 0.82. With- out angle of attack (or for symmetric flow), a stronger normal shock wave occurs about at the. middle of the body surface (M, = 0.85). Then

the shock wave moves further backward even to the trailing edge (Fig. 6 and Fig. 7) with the increase of Mach number and the shock waves become oblique. For. iL& = 0.95, there is even a shock wave interaction behind the trailing edge. When Mach number is slightly greater than 1, a compression wave at first occurs in front of the leading edge (IL!& = 1.05), while the interaction of shock wave at trailing edge region is not so evident as that of it& = 0.95.

1 I 1.5

- 1

- 0.5

-0Y

- -0.5

- -1

I I -1.5 -1.5 -1 -0.5 0 0.5 1 1.5 2

Figure 3: Mi = 0.75

I I i 'r 1.5 / ‘3,

- 1

- 0.5

- OY

- -0.5

- -1

\ ./ . -1.5

-1.5 -1 -0.5 0 0.5 1 1.5 2

Figure 4: M, = 0.82

1.5

1

0.5

OY

-0.5

-1'

-1.5

Figure 5: Mm = 0.85

We also make some computations for super-

3


:

1.5

1

0.5

OU

-0.5

-1

-1.5 -1.5 -1 -0.5 0 0.5 1 1.5 2

=/I

Figure 6:-M, = 0.90

1.5

1

0.5

OY

-0.5

-1

-1.5 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 k -1.5

-1 -0.5 0 0.5 1 1.5 2 2.5

a/'

Figure 7: Mm = 0.95 Figure 10: Mm =‘2.50

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2.5

Figure 8: AL = 1.05

sonic region. For M, = 1.5, there is an oblique shock wave but still detached due to the blunt head of NACA 0012 together with a weak trailing shock wave at the back region. Even for Mm = 2.5, the shock wave in front’ of the head is heavily inclined but still detached (Fig. 28).

Comparison with 1st Order TVD Scheme

In order to show’ the resolution to shock wave for the second-order accurate TVD scheme, a comparison is made with the first-order accurate

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

=I’

Figure 9: Mm = 1.50

TVD scheme (that is, let limiter g& terms zero). It shows in Fig. 11 that there is a clear difference of the distribution of the pressure coefficient C, along the body surface for the 1st and 2nd order accurate TVD schemes. This is more clearly described in Fig. 12 and Fig. 13 where the local Mach number contours are illustrated during capturing shock waves. From the .Mach number contouis, the first-order accurate scheme hasn’t compressed the shock wave enough, hence making the shock.wave heavily smeared.

Figure 11: x/Z Y C, for 1st and 2nd order accurate TVD schemes

4


IIIIJIIJIII -0.05 -0.2 0 0.2 0.4 0.6 0.6 1 1.2 1.4 1.6 1.6

Figure 12: iI& = 0.85 (2nd order)

-0.05 -0.2 0 0.2 0.4 0.6 0.6 1 1.2 1.4 1.6 1.6

=/I

Figure 13: IL& = 0.85 (1st order)

Capturing of Shock Wave for Different Angles of Attack

The computations are made for different angles of attack: Q = 1”,3”, 10” with Mm = 0.85, Re = 1E5(with Mach number contours in Fig. 14 to Fig. 16). At this case, the shock wave of Q = lo on the upper and lower surfaces moves some further apart than that of (Y = 3”, and there is a shock wave interaction on upper surface for LY = 10” (Fig. 16). For the laminar assumption adopted here, the case of a! = 10” might be an unsteady state because this case should be turbulent. For this reason, the iteration history of the Q = 10” case becomes a periodical oscillation curve after some commutation steas, as shown in Fig. 17.

Process to Steady-State Solution

Because the time-dependent method is used to get the steady-state solution, we provide some middle-stages of C, distribution for ‘M, = 0.85,~ = 0” case shown in Fig. 18. During the convergence process, a. general expansion- compression wave fans at first forms in the field (Fig. 19), then an almost fixed* shock wave is

1.5

1

0.5

OY

-0.5

-1

-1.5 -1.5 -1 -0.5 0 0.5 1 1.5 2

=/I

Figure 14: Mm = 0.85, CY = 1”

1.5

0.5

OY

-0.5

-1

-1.5

Figure 15: Mm = 0.85, o = 3“

’ . . . . . . . . . . . . . . . . . . . . .I

I I I -1.5 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

Figure 16: Ma = 0.85, CY = 10”

formed rapidly at the back half-surface region (here about 0.5 to 0.6 of the body length region, as shown in Fig. ZO), but it needs quite a long process to converge to the final fixed po- sition. However, this process is much easier for the first-order accurate scheme because of its implicit large artificial viscosity. From this point, the resolution to shock wave and the convergence to steady state solution of the schemes are just a contradictory, and more pronouncing in tran-

5

(c)l999 American Institute of Aeronau+.zs & Astronautics or published with permission of author(s) arid/or author(s)’ sponsoring organization.

Figure 17: Iteration history

.12 I 0 0.1 0.2 0.3 0.1 0.5 073 0.7 0.1 0.0 t

r/l

Figure -18: C, distributions of middle process

I 1.5

/ - 1

I ‘. -1.5

-!.5 -1 -0.5 0 0.5 1 1.5 2 2.5

=/'

Figure 19: Iteration steps=500 A& = 0.85

- 0.5

- -0.5

-1.5 -1 -0.5 0 0.5 1 1.5 2' 2.5

Figke 20: Iteration steps=lOOQ A&, ‘= 0.85

sonic flows.

The Interaction of Shock Waves

As mentioned above, there are interactions of shock waves for the cases it&, = 0.95, a = 0” and M = 0.85, (Y = 10”. For better understanding, their local pressure contours and density contours are given below.

For the case of it&, = 0.95,a = 0” (Fig. 21 and Fig. 22), the shock wave appears at the trailing edge of the body since the whole field almost becomes supersonic and a normal compression wave almost occurs in front of the body head. After this shock wave, there is a second compression at the back region and a compression wave forms because of the deflection of the flow. This shock wave and the compression wave then forms a stronger shock at the back region symmetrically on the upper and lower sides.

I I I I , 1.2

I I I I I 1 -0.2 -1 -0.5 0 0.5 1 1.5 2 2.5

21'

Figure 21: pressure contour M, = 0.95

I I I I , 1 1.2

6

I I ' -0.2 -1 -0.5 0 0.5 1 1.5 2 2.5

Figure 22: density contour Moo = 0.95

For the case of I&, = 0.85,a = 10” (Fig. 23 and 24), there is a more evident shock wave interaction. Furthermore, the flow field near the body surface becomes turbulent. For the flow case in Fig. 26, there is an obvious flow separation with vortexes and reattachment formed just after the


appearance of the first shock wave on the body

@

surface. This flow separation then makes flow de- fleeted at the trailing edge region, forming a second shock wave. Consequently, these two shock waves composite into a stronger shock wave on the upper side.

- 1.2

-1

- 0.8

- o&J

- a.4

- 0.2

- 0

1 I 8 I 1 -0.2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

./I

Figure 23: pressure contour i&, = 0.85

I I I I 1 -0.2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

Figure 24: density contour A& = 0.85

Velocity Flow near Body Surface

When there is no angle of attack, the flow keeps a good laminar flow condition at this numerical test for both transonic and supersonic regions. When there is a low angle of attack, the flow separation is not evident (e.g. Mm = 0.85, a! = 1” and 3’) so that the laminar assumption is still effective (Fig. 25). With a higher angle of attack (a: = lo”), the flow becomes tur- buIent (Fig. 26), there is not onIy a flow separation occurred on the upper surface just after the appearance of the first shock wave, but also a stronger disturbance flow appeared at the trailing region from the lower surface, forming a

7

>

Figure 25: A&, “= 0.85, LL = 3”

-\ ‘1 ’ \r ’ \ ’ \ ’ \ 1 . / . ‘fl - s- 1.3

Figure 26: A&, = 0.85, a = 10”

second shock wave and a wake region.

The Shock Wave in front of Blunt Body at Supersonic Flow

In order to show the shock wave situations in front of the blunt body, we give the local Mach number contours in Fig. 27 and Fig. 28. De- spite that the Mach number is rather high, there is still a detached shock wave in front of the body head with a small subsonic region between them. Clearly, the distance between the shock wave and the body head decreases with the increase of Mach number. Furthermore, because of the higher resolution to the shock wave for the TVD scheme, it is much easier for the supersonic time- dependent computation to ‘converge to a steady state solution.

5 Remarks Here we use a second-order non-MUSCL TVD

scheme to have made the computations for both transonic and supersonic regions. The shock wave is successfully captured with a good resolution. However, only the laminar assumption is applied here, so it is necessary to extend this

(c)1999 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

I I I 1 -0.4

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

z/I

Figure 27: Mm = 1.50,

1 0.25.

0.2

0.15

0.1

0.05

OU

-0.05

-0.1

-0.15

-0.2 . . I I I I * I I I1 IS I

-0.25 -0.2 -0.15 -0.25

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

./I

Figure 28: M, = 2.50

application with a suitable turbulent modelling and computing.

Finally, the authors thank the China Scholar- ship Council for its generous support of research. Also, the Mathematics Department of Manch- ester University has shown great understanding in allowing us the necessary time to carry out the computations.

8


PI

PI

[31

PI

PI

PI

PI

PI

PO1

WI

References

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[American Institute of Aeronautics and Astronautics 9th International Space Planes and Hypersonic Systems and Technologies Conference - Norfolk,VA,U.S.A. (01 November 1999 - 05 November