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An immersed-boundary method for compressible viscous flows

P. De Palma, M.D. de Tullio, G. Pascazio, M. Napolitano *

DIMeG, Sez. Macchine ed Energetica, and CEMeC, Politecnico di Bari, Via Re David 200, 70125 Bari, Italy

Available online 20 February 2006

Abstract

This paper combines a state-of-the-art method for solving the preconditioned compressible NavierStokes equations accurately andefficiently for a wide range of the Mach number with an immersed-boundary approach which allows one to use Cartesian grids for arbi-trarily complex geometries. The method is validated versus well documented test problems for a wide range of the Reynolds and Machnumbers. The numerical results demonstrate the efficiency and versatility of the proposed approach as well as its accuracy, from incom-pressible to supersonic flow conditions, for moderate values of the Reynolds number. Further improvements, obtained via local gridrefinement or non-linear wall functions, can render the proposed approach a formidable tool for studying complex three-dimensionalflows of industrial interest. 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Many fluid dynamic problems of engineering interest

exhibit flow regions with very different Mach numbers,which render their numerical simulation very difficult. Infact, due to the different nature of the physical phenomenaassociated with flows at different Mach numbers, usually asingle numerical method performs accurately and effi-ciently within a limited range of the Mach number. More-over, the presence of complex and/or moving boundariesusually requires time consuming body-fitted grid genera-tions. The aim of the present work is to remedy both ofthe aforementioned difficulties, by combining a state-of-the-art method for solving the preconditioned com-pressible NavierStokes equations accurately and effi-

ciently for a wide range of the Mach number with animmersed-boundary (IB) approach which allows to useCartesian grids for arbitrarily complex geometries. Con-cerning the preconditioning of the governing equations,the residual of the compressible NavierStokes or Rey-nolds-averaged NavierStokes (RANS) equations is

premultiplied by a suitable matrix which uniforms thewave propagation speeds, thus greatly enhancing the accu-racy and efficiency of the compressible flow solver when

applied to low-Mach-number flows. Such a precondition-ing technique, originally designed for steady flows [1,2],has been extended to the unsteady ones using a dual-time-stepping (DTS) technique with a three-level back-ward discretization of the time derivative, in conjunctionwith a third-order-accurate finite volume method basedon flux-vector splitting for the convective terms [3,4].The IB technique, originally designed for incompressibleflows [5,6], allows the body surface to cut the computa-tional cells, so that a simple Cartesian grid can beemployed, independently of the complexity of the consid-ered geometry. In this work, the IB technique has been

extended to the preconditioned compressible NavierStokes and RANS equations to provide an accurate,efficient and versatile tool for studying complex three-dimensional flows of industrial interest. Here the proposedmethod is validated versus two-dimensional flows for avery wide range of the Reynolds and Mach numbers. Inthe following sections, a brief review of the governingequations and their solution technique is given at first,then, the IB technique is described and, finally, resultsare provided and compared versus numerical and experi-mental ones available in the literature.

0045-7930/$ - see front matter 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compfluid.2006.01.004

* Corresponding author. Tel.: +390805963464; fax: +390805963411.E-mail addresses: [email protected] (P. De Palma), detullio@

imedado.poliba.it (M.D. de Tullio), [email protected] (G. Pascazio),[email protected] (M. Napolitano).

www.elsevier.com/locate/compfluid

Computers & Fluids 35 (2006) 693702
mailto:[email protected]:detullio@%20imedado.poliba.itmailto:detullio@%20imedado.poliba.itmailto:[email protected]:[email protected]:[email protected]:[email protected]:detullio@%20imedado.poliba.itmailto:detullio@%20imedado.poliba.itmailto:[email protected]

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2. Governing equations and numerical method

The Reynolds-averaged NavierStokes (RANS) equa-tions, written in terms of Favre mass-averaged quantitiesand using the standard kx turbulence model, can bewritten as follows:

oqot ooxj

quj 0; 1

oqui

ot

o

oxjqujui

op

oxiosji

oxj; 2

oqU

ot

o

oxjqujH

o

oxjuisijlr

ltok

oxjqj

; 3

oqk

ot

o

oxjqujk sij

oui

oxjbqxk

o

oxjlrlt

ok

oxj

; 4

oqx

ot

o

oxjqujx

cx

ksij

oui

oxjbqx2

o

oxjlrlt

ox

oxj

. 5

In the equations above, U and H are the total energy andenthalpy comprehensive of the turbulent kinetic energy,k; the eddy viscosity, lt, is defined in terms of k and ofthe specific dissipation rate, x, according to the kx turbu-lence model of Wilcox [7], namely:

lt c qk

x. 6

Moreover, sij indicate the sum of the molecular andReynolds (sij) stress tensor components. According to theBoussinesq approximation, one has:

sij l ltoui

oxjouj

oxi

2

3

ouk

oxkdij

2

3

qkdij. 7

Finally, the heat flux vector components, qj, are given as

qj l

Pr

ltPrt

oh

oxj; 8

where Pr = 0.71 and Prt = 1 are the laminar and turbulentPrandtl numbers, respectively. The Sutherland law is usedto compute the molecular viscosity coefficient. Finally,the standard coefficients of the turbulence kx model areused [7], namely:

b 3

40; b

9

100; c

5

9; c 1; r r

1

2.

The numerical method employed to solve the RANS equa-tions is described in the following with reference to the two-dimensional case, for simplicity. The system of equations iswritten either in Cartesian or generalized curvilinear coor-dinates, (n,g); a pseudo-time derivative is added to the left-hand-side in order to use a time marching approach forboth steady state and unsteady problems; the precondition-ing matrix, C, proposed in [1,2] is finally used to premulti-ply the pseudo-time derivative in order to improveefficiency. The final system reads

CoQv

os

oQ

ot

oE

on

oF

og

oEv

on

oFv

og

D; 9

where Q is the conservative variable vector, E, F, and Ev,Fv indicate the inviscid and viscous fluxes, respectively, Dis the vector of the source terms for the turbulence equa-tions, and Qv = (p, u, v, T, k,x)

T is the primitive variablevector, which is related to Q by the Jacobian P = oQ/oQv. Discretizing equation (9) by an Euler implicit scheme

in the pseudo-time and approximating the physical-timederivative by second-order-accurate three-point backwarddifferences, the following equation in delta form isobtained:

C3

2

Ds

DtP Ds

o

onAv Rnn

o

on Rng

o

og

Dso

ogBv Rgg

o

og Rgn

o

on

DQv

Ds3Qr 4Qn Qn1

2DtRr

; 10

where r andDs indicate the pseudo-time level and step, nand Dt indicate the physical-time level and step, Av = oE/

oQv, Bv = oF/oQv, Rij are the viscous coefficient matrices[8], and the matrix C is evaluated as proposed in [3,4].The residual is given as

Rr

oErErv

onoFrFrv

ogDr; 11

and the delta unknowns to be annihilated at every pseudo-time level are

DQv Qr1v Q

rv. 12

The left-hand-side (LHS) of Eq. (10) is modified to improve

the efficiency of the method, without affecting the residual,that is, the physical solution. Firstly, the non-orthogonalviscous coefficient matrices, Rng and Rgn, are neglected,and the remaining ones are approximated by the corre-sponding spectral radii multiplied times the identity matrix,Rnn = RnI and Rgg = RgI; then, as proposed in [8], thepseudo- and physical-time terms are grouped together intoa new term S,

S C3

2

Ds

DtP; 13

which is factored out of the LHS in Eq. (10), yielding

S I DsS1 oon

Av RnIo

on

DsS1 o

ogBv RgI

o

og

DQv Ds3Qr 4Qn Qn1

2DtRr

. 14

In order to solve the resulting linear system, the diago-nalization procedure of Pulliam and Chausee [9] is firstlyapplied, so that the matrices S1Av and S

1Bv can bewritten as

S1Av MnKnM1n ; S

1Bv MgKgM1g ; 15

where Mn, Mg are the right-eigenvector matrices, M1n , M

1g

are the left-eigenvector matrices; and Kn and Kg are diago-

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nal matrices containing the eigenvalues of S1Av andS1Bv, respectively; then, the LHS of Eq. (14) is factorized,

SMn I Dso

onKn RnI

o

on

M1n Mg

I Dso

og

Kg RgIo

og

M1g DQv Ds

3Qr 4Qn Qn1

2DtRr

; 16

and solved by a standard scalar alternating direction implicitprocedure [4]. A cell-centred finite volume space discret-ization is used on a multi-block structured mesh. A third-order-accurate flux vector splitting scheme is employed todiscretize the convective terms, the minmod limiter beingapplied in the presence of shocks, whereas the viscousterms are discretized by second-order-accurate central dif-ferences. Further details of the method can be found inRef. [8], which is available on the web [10], together with

the code developed at the Pennsylvania State University.

3. Immersed-boundary technique

The immersed boundary (IB) technique used in thiswork is based on that proposed in [5,6]. In a preliminarystep, the geometry under consideration, which is describedby a closed polygon in two dimensions (a closed surface inthree dimensions), is overlapped onto a Cartesian (non-uniform) grid. Using the ray tracing technique based onthe geometrical algorithms reported in [11], the computa-tional cells occupied entirely by the flow are tagged as inter-

nalcells; those whose centres lie within the immersed bodyare tagged as external cells; the remaining ones are finallytagged as interface cells. The main feature of the IB tech-nique is the evaluation of the unknowns at the centres ofthe interface cells. Here, the direct forcingmethod proposedby Mohd-Yusof [12] is employed. For each interface cell,the shortest Cartesian distance between the cell centreand the solid wall is determined. Along the correspondingdirection, see Fig. 1, the variables at the centre of the inter-face cell (point P) are linearly interpolated between the val-ues to be imposed at the boundary point (point B) and thecomputed values at the neighbouring internal-cell centre(point A), except for the pressure, whose value at point Pis set equal to that at point A, which amounts to imposinga first-order-accurate homogeneous Neumann conditionfor the pressure. In the present work, Dirichlet boundaryconditions are imposed to u1 and u2 (the two Cartesianvelocity components), T (the temperature) and, for turbu-lent flows, kand x. As shown in [5], such an approach pro-vides an essentially second-order-accurate solution. It isnoteworthy that, for high Reynolds number flows, a non-linear interpolation procedure which utilizes adaptive wallfunctions is needed to obtain accurate results. Needless tosay, the governing equations are solved at all internal cellcentres, whereas all unknowns are set to zero at external

ones.

4. Results

The proposed methodology has been applied to com-pute two-dimensional steady and unsteady flows, for awide range of the Reynolds and Mach numbers.

4.1. Incompressible flow past a circular cylinder

The two-dimensional incompressible flow past a circularcylinder has been considered at first to test both the precon-

ditioning strategy and the immersed-boundary method ver-sus steady as well as unsteady flows at very low Machnumbers. A single value of the free-stream Mach number,M1 = 0.03, and four values of the Reynolds number,based on the cylinder diameter, D, the free-stream velocity,U1

, and kinematic viscosity, m1

, namely, 20, 40, 100, and200, have been considered; the first two cases correspond tosteady flow regimes and the last ones to unsteady ones. Arectangular computational domain has been used withthe inlet and outlet vertical boundaries located at

- Interface cell

- External cell

- Internal cell

^

^

immersedboundary

A P B

Computed

Interpolated

Fig. 1. Schematic representation of the interpolation scheme for the flowvariable at interface cells.

X

Y

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Fig. 2. Local view of the grid.

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xi = 10D and xo = 40D and the two horizontal bound-aries located at yf= 20D, respectively, the origin coincid-ing with the centre of the cylinder. Standard characteristic

boundary conditions have been imposed at inlet and outletpoints, whereas free-shear wall boundary conditions areimposed at the points on the far-field horizontal bound-aries. Computations have been performed using a struc-tured, non-uniform, single-block grid with 244 168 cells.A local view of the mesh is given in Fig. 2. Fig. 3 showsthe steady streamlines corresponding to Re = 20 andRe = 40, respectively. Finally, the computed geometricalproperties of the symmetrical vortices, as defined inFig. 4, and the drag coefficient, CD, are provided in Tables1 and 2, together with the corresponding experimental[13,14] and numerical [1517] results available in the litera-

ture. The agreement is quite satisfactory. For both compu-tations, the steady solver has been used, which annihilatesthe physical time derivative and iterates till the steadyresidual is reduced to 106. Convergence is achieved withinabout 18,220 iterations, corresponding to 5160 CPU sec-onds on a single processor Pentium IV (2.6 GHz). Con-cerning the Re = 100 and Re = 200 unsteady flowcomputations, the non-dimensional physical time step hasbeen set equal to 0.03, which corresponds to about 200

steps per period. About 200 inner iterations are needed toreduce the unsteady residual to 106 at every physical timestep, corresponding to about 63 CPU seconds on the afore-mentioned processor. Two snapshots of the vorticity con-tours are given in Fig. 5; in both cases, the lift and dragcoefficients have regular sinusoidal behavior in time, asshown in Fig. 6 for Re = 200. Finally, the computed Strou-hal number based on the shedding frequency, f (St = fD/U1

), as well as the drag and lift coefficients are given inTables 3 and 4 together with the experimental [18] andnumerical [17,19,20] results available in the literature. Alsofor these unsteady flow cases, a very good agreement isobtained.

4.2. Unsteady flow past a heated circular cylinder

The unsteady very low-Mach number flow past a heatedcircular cylinder has been chosen in order to validate theproposed method for a flow in which the energy equation

plays a significant role, insofar as experimental [21,22]

X

Y

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-1.5

-1

-0.5

0

0.5

1

1.5

X

Y

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-1.5

-1

-0.5

0

0.5

1

1.5

Fig. 3. Steady streamlines for Re = 20 (left) and Re = 40 (right).

X

Y

0 1 2 3

0

1

L

a

b

Fig. 4. Definitions of the relevant geometrical parameters of the

symmetric separation region behind the cylinder.

Table 1Steady flow past a circular cylinder at Re = 20

L a b h CDFornberg [15] 0.91 45.7 2.00Dennis and Chang [16] 0.94 43.7 2.05Coutanceau and Bouard [13] 0.93 0.33 0.46 45.0 Tritton [14] 2.09Linnick and Fasel [17] 0.93 0.36 0.43 43.5 2.06Present 0.93 0.36 0.43 44.6 2.05

Table 2Steady flow past a circular cylinder at Re = 40

L a b h CD

Fornberg [15] 2.24 55.6 1.50Dennis and Chang [16] 2.35 53.8 1.52

Coutanceau and Bouard [13] 2.13 0.76 0.59 53.8 Tritton [14] 1.59Linnick and Fasel [17] 2.28 0.72 0.60 53.6 1.54Present 2.28 0.72 0.60 53.8 1.55


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and numerical [23] investigations indicate that the temper-ature field has a significant influence on the flow pattern,especially when the ratio between the cylinder wall temper-

ature, Tw, and the free-stream one, T1 (Tw = Tw/T1)

exceeds 1.1. In particular, it has been found that, for agiven Re1, the vortex shedding frequency, f, i.e., the Strou-hal number, St, decreases for increasing values ofTw. Fur-thermore, by defining an effective Reynolds number, Reeff,in terms of the kinematic viscosity corresponding to themaximum temperature in the wake, Teff (which happensto be at the centre of the first vortex shed by the cylinder,the curves St versus Reeff, obtained for different values of

T

w

in the range 16

T

w6

2, collapse into a single curve;see Ref. [21], where the following empirical correlationwas proposed and employed to compute Teff:

Teff T1 0:28Tw T1. 17

In the present work, the same rectangular domain em-ployed in the previous test-case has been used and discret-ized with a much finer (non-uniform) multi-block Cartesiangrid with 142,200 cells and 12 blocks, see Fig. 7, where alocal view of all blocks is provided (only one every fivegrid-lines is plotted). Computations have been performedon a cluster with 12 processors for M

1= 0.01 and several

values of Re and Tw. The non-dimensional time step has

been set equal to 0.012, which corresponds to about 500

X

Y

-2 0 2 4 6 8 10 12-5

-4

-3

-2

-1

0

1

2

3

4

5

X

Y

-2 0 2 4 6 8 10 12-5

-4

-3

-2

-1

0

1

2

3

4

5

Fig. 5. Snapshot of the vorticity contours for Re = 100 (left) and Re = 200 (right).

Time (s)

C

8 9 10 11 12-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

L

Time (s)

C

8 9 10 11 121.28

1.3

1.32

1.34

1.36

1.38

1.4

D

Fig. 6. Time history of the lift and drag coefficient for Re = 200.

Table 3Unsteady flow past a circular cylinder at Re = 100

St CD CL

Berger and Willie [18] 0.160.17 Liu et al. [19] 0.165 1.35 0.012 0.339Linnick and Fasel [17] 0.166 1.34 0.009 0.333

Present 0.163 1.32 0.01 0.331

Table 4Unsteady flow past a circular cylinder at Re = 200

St CD CL

Berger and Willie [18] 0.180.19 Belov et al. [20] 0.193 1.19 0.042 0.64Rogers and Kwak,

reported in [20]0.185 1.23 0.050 0.65

Miyake et al., reported in [20] 0.196 1.34 0.043 0.67Liu et al. [19] 0.192 1.31 0.049 0.69Linnick and Fasel [17] 0.197 1.34 0.044 0.69Present 0.190 1.34 0.045 0.68


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steps per period. About 200 inner iterations are needed toreduce the unsteady residual to 106 at every physical timestep, corresponding to about 55 CPU seconds. The com-puted values of St for Re1 = 80, 100, 120, 140 andTw = 1, 1.1, 1.5, 1.8 are provided in Fig. 8, together withthe experimental results of [21]: a very good agreement isobtained. Then, numerical results have been obtained fora wider range of Tw, for Re

1= 140 and 260. The com-

puted values of St versus Reeff are given in Fig. 9, togetherwith those already shown in Fig. 8. A snapshot of the tem-perature contours for Tw = 2 and Re1 = 260 is provided inFig. 10, for completeness. Finally, Table 5 provides the val-

ues of St and Teff computed either at the centre of the firstshed vortex, T1

eff, see Fig. 10 or using Eq. (17), T2eff. The

present results confirm the experimental data of [21], to-gether with the validity of Eq. (17) in the range 1 6 Tw 6 2;moreover, they show that for Re

1= 140 and Tw = 3.5 the

flow becomes steady and for Re1 = 260 and TwP 2.5 the

values ofSt do not fit the universal curve St(Reeff), prob-ably because a change in the physical nature of the unstea-dy phenomenon occurs at 140 < Re < 260. Notice that forRe1

= 260 the values corresponding toReeff> 160 are notplotted insofar as they lie outside the range of laminar peri-odic wakes (they fall in the A-mode transition region forthe case of unheated cylinders). Finally, for the twoRe1 = 260 cases with T

w = 1.6 and Tw = 3, the computedvalues of Teff are in good agreement with the experimentaldata of [22] for the maximum temperature in the wake,namely, Tmax = 343.5 K for T

w = 1.61 and Tmax = 484.1 Kfor Tw = 2.98.

4.3. Supersonic flow past an NACA0012 airfoil

In order to test the proposed methodology versus a welldocumented viscous flow at high Mach number, the lami-nar supersonic flow past an NACA0012 airfoil withM1 = 2, a = 10 and Re1 = 1000 has been considered[24]. Three grids with 1252, 2502, and 5002 cells have beenused to discretize the computational domain [8c; 9c] [8c; 8c], c being the chord-length of the airfoil, whose

X

Y

-0.002 -0.001 0 0.001 0.002 0.003 0.004 0.005 0.006

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

Fig. 7. Local view of the multi-block grid.

Re

St

60 70 80 90 100 110 120 130 140 150 1600.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

- T*=1.0

- T*=1.5- T*=1.1

- T*=1.8

Fig. 8. Strouhal number versus Reynolds number for flow past a heatedcylinder: comparison between experimental (open symbols) and numerical

(solid symbols) data.

*

+

*

*+

Re

St

50 60 70 80 90 100 110 120 130 140 150 1600.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

- T*=1.5

- T*=1.1- T*=1.4

eff

- T*=1.0

- T*=1.6

- T*=3.0

- T*=1.8- T*=2.0- T*=2.5

- T*=3.5

Fig. 9. Strouhal number versus effective Reynolds number for flow past aheated cylinder: comparison between experimental data (open circles) for

Tw = 1 and numerical results for Tw > 1.

X

Y

0 0.002 0.004 0.006 0.008 0.010 0.012

-0.004

-0.002

0

0.002

Teff

Fig. 10. Snapshot of the temperature contours for Tw = 2.0 and Re1

=

260: the point at which Teff has been evaluated is indicated.


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leading edge is located at the origin; Fig. 11 shows a localview of the finest mesh (only one every five grid lines isplotted), which is partitioned into 18 blocks, for parallelcomputing. A benchmark solution has also been obtainedfor comparison purposes, using the present approach ona very fine body-fitted grid (87,500 cells with 12 blocks).The pressure coefficient distributions along the profile aregiven in Fig. 12 where the finest grid solution, which coin-cides with the aforementioned benchmark one, within plot-ting accuracy, is clearly seen to be grid-converged. The liftand drag coefficients obtained on the three grids are equalto 0.3296, 0.3335, 0.3353, and 0.2448, 0.2485, 0.2514,respectively, which tend towards the values of 0.3400 and0.2515, obtained from the aforementioned benchmarksolution, as the mesh is refined. Finally, the Mach numbercontours computed on the finest grid are provided inFig. 13 showing that the shock is computed monotonically.Using the steady solver, reducing the residual to 106 onthe finest grid requires about 105 iterations, correspondingto 2 105 CPU seconds on a single processor Pentium IV(2.6 GHz).

Two comments are in order. All of the lift and drag coef-ficients obtained from computation using the IB approachand Cartesian grids are computed performing a momen-

tum balance of the fluid comprised within a rectangle sur-

rounding the body. It has been verified that by varying therectangle dimensions, the computed results vary less thanone-tenth of 1%.

In the heated cylinder case, the block containing the cyl-inder contains about one-fifth of the total number of cells;in the present flow case, the 18 blocks all contain about the

Table 5Temperatures in K

Re = 140 Re = 260

Tw T1eff T

2eff St

(1) Tw T1eff T

2eff St

(1)

1.1 303.5 303.3 0.178 1.1 303.7 303.3 0.1911.4 329.6 328.0 0.170 1.4 329.4 328.0 0.189

1.6 345.1 344.6 0.167 1.6 346.2 344.6 0.1881.8 364.2 361.1 0.163 1.8 364.4 361.1 0.1872.0 382.2 377.6 0.161 2.0 381.5 377.6 0.1852.5 422.5 418.9 0.150 2.5 423.9 418.9 0.1833.0 476.2 460.2 0.140 3.0 475.0 460.2 0.1773.5 501.5 3.5 520.0 501.5 0.171

(1) Present computations, (2) Eq. (17).

X

Y

-1 -0.5 0 0.5 1 1.5 2

-1

-0.5

0

0.5

1

Fig. 11. Local view of the mesh for the flow past an NACA0012 airfoil.x/c

Pressurecoefficient

0 0.25 0.5 0.75 1-2

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

500 x 500

250 x 250

125 x 125

Fig. 12. Pressure coefficient distributions along the NACA0012 profile.

X

Y

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-1

-0.5

0

0.5

1

1.5

2

Fig. 13. Mach number contours (DM= 0.1).


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same number of cells. Accordingly, the CPU time per iter-ation and grid point required by the parallel computationsare five and 18 times lower than those required by thesingle processor calculations, respectively.

4.4. Supersonic flow past a circular cylinder

Finally, supersonic turbulent steady flows past a circularcylinder at Re

1= 2 105 and M

1= 1.3 and 1.7, have

been considered as formidable test-cases for the proposedIB method. The inlet values of the turbulence kineticenergy and specific dissipation rate are k=U21 0:0009and xD/U1 = 40, respectively. For the considered valuesof M1, a bow shock is formed upstream of the cylinder;the subsonic flow at the front part close to the wall acceler-ates along the surface forming a supersonic-flow region,enveloping the subsonic recirculation region behind the cyl-inder, and two symmetric tail shocks are formed at the endof the separation region. Results have been obtained using

a rectangular computational domain [8D; 9D] [8D;8D], D being the diameter of the cylinder centred at the ori-gin. Two non-uniform Cartesian grids have been employedfor the IB computations, the first one being a single-blockgrid with 430 200 cells and the second one having 10blocks and 949,946 cells; the average values of y+ corre-sponding to the first grid point away from the cylinderare about 50 and 10, respectively. A partial view of the finergrid is shown in Fig. 14, where only one every 10 grid linesis plotted and the thick horizontal lines delimitate eachblock. Furthermore, two computations have been per-formed using body-fitted grids having 12 blocks and

99,913 and 188,423 cells (y+

= 0.1 and 0.05), for compari-son. Figs. 15 and 16 provide a local view of the Mach num-ber contours around the cylinder obtained using the IBmethod on the finer grid. It is noteworthy that the lengthof the recirculation region decreases as the Mach numberincreases. The computed positions of the separation point,hs, h being the clockwise angle measured from the leading

edge, are given in Table 6, together with the correspondingexperimental data of [25]; moreover, the computed and

experimental drag coefficients are reported in Table 7. Allnumerical results agree reasonably well with the experimen-tal data; in particular, also the coarse-grid results are satis-factory in spite of the inadequate resolution of theboundary layer region. Finally, the computed pressure

X

Y

-3 -2 -1 0 1 2 3

-2

-1

0

1

2

Fig. 14. Local view of the mesh for the supersonic flow past a cylinder.

X

Y

-4 -2 0 2 4 60

2

4

6

Fig. 15. Local view of the Mach number contours for M1

= 1.3 (DM=0.08).

X

Y

-4 -2 0 2 4 60

2

4

6

Fig. 16. Local view of the Mach number contours for M1

= 1.7

(D

M= 0.08).

Table 6Supersonic flow past a circular cylinder: separation-point angle, hs

Immersed boundary Body-fitted Ref.[25]

Coarse-grid Fine-grid Coarse-grid Fine-grid

M1

= 1.3 107 105 102 103 103M1

= 1.7 118 111 111 112 112

Table 7Supersonic flow past a circular cylinder: drag coefficient, CD

Immersed boundary Body-fitted Ref.[25]Coarse-grid Fine-grid Coarse-grid Fine-grid

M1

= 1.3 1.46 1.44 1.45 1.45 1.48

M1= 1.7 1.38 1.39 1.40 1.40 1.43


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coefficient distributions along the surface of the cylinderare provided in Figs. 17 and 18 together with the experi-mental data of [25]. The two body-fitted solutions aregrid-converged, insofar as they coincide within plottingaccuracy. The coarser-grid IB solution is not satisfactory,especially in the aft-body separation region, whereas thefiner-grid solution is smoother and tends towards grid con-vergence. On the other hand, the IB approach, at its pres-ent state, is not yet competitive for computing highReynolds number flows. In this respect, it is noteworthythat for the finer-grid solution the steady-state solverrequires about 150,000 iterations to reduce the residual to106, corresponding to 137,700 CPU seconds using 10 Pen-tium IV (2.6 GHz) processors. However, the more accuratecoarse-grid body-fitted solution requires about 50,000 iter-ations to reduce the residual to 106, corresponding to54,000 CPU seconds using a single processor. A final com-ment on the experimental results is warranted. In Ref. [25],the authors provide numerical solutions which coincidewithin plotting accuracy with the present body-fitted ones.The minor discrepancies between the numerical solutionsand the experimental results are thus believed to be due

to three-dimensional or wall effects in the experiments,

rather than to inadequate turbulence modelling, insofaras they are equally important in both the attached andseparated flow regions.

5. Conclusions

A state-of-the-art method for solving the preconditionedcompressible NavierStokes equations accurately and effi-ciently for a wide range of the Mach number is combinedwith an immersed-boundary approach which allows touse Cartesian grids for arbitrarily complex geometries.The methodology has been applied to compute steadyand unsteady flows past circular cylinders and anNACA0012 airfoil for a wide range of the Reynolds andMach numbers demonstrating its versatility as well as itsaccuracy for moderate values of the Reynolds number.For high-Reynolds number flows the proposed approachis not yet competitive, insofar as it requires huge numbers

of grid points to provide accurate and reliable solutions. Alocal refinement strategy or non-linear wall-laws, which canmimic an accurate resolution of the viscous sub-layer, areneeded to obtain a state-of-the-art tool for investigatingthree-dimensional flows of industrial interest. Bothapproaches are currently under investigation by the presentas well as other researchers.

Acknowledgments

This research has been supported by MIUR and Politec-nico di Bari, grants Cofin2003 and Cofinlab2000. The

authors are grateful to their colleagues of PennsylvaniaState University, who have developed and made availablethe basic computer code, as well as to G. Iaccarino andR. Verzicco for valuable suggestions and discussions onthe immersed boundary method.

References

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Cp

0 20 40 60 80 100 120 140 160 180-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Immersed Boundary (fine grid)Immersed Boundary (coarse grid)Experimental

Body Fitted (fine grid)Body Fitted (coarse grid)

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Fig. 18. Pressure coefficient distribution along the surface of the cylinder:comparison between experimental and numerical results for M1

= 1.7.


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