Inviscid flows over a cylinder

Comput. Methods Appl. Mech. Engrg. 193 (2004) 1981–1995

www.elsevier.com/locate/cma

Inviscid flows over a cylinder

Mohamed Hafez *, Essam Wahba

Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA 95616, USA

Received 9 June 2003; received in revised form 16 October 2003; accepted 8 December 2003

Abstract

In this paper, we simulate steady inviscid flows over a cylinder using potential and stream functions, including

entropy and vorticity corrections for incompressible, subsonic, transonic and supersonic flows. The present hierarchical

formulation is equivalent to Euler equations satisfying conservation of mass, momentum and energy. Standard

numerical schemes and iterative algorithms are used for the mixed type potential (or stream function) equation and for

the convection equations of entropy and vorticity augmented with the proper artificial dissipation necessary for stability

of computations. Typical results for a cylinder in uniform and shear flows are presented and the merits of the present

approach are briefly discussed.

� 2004 Elsevier B.V. All rights reserved.

Keyword: Inviscid flows

1. Introduction

A cylinder represents a very basic geometry and different types of flows over a cylinder have beenexperimentally and theoretically studied over the years. Recently, a book has been published on low speed

flows around a circular cylinder [1]. Interesting experimental results can be found in Album of Fluid

Motion by van Dyke [2], see also [3]. The analytical solution for incompressible inviscid flow over a cylinder

resulted in D�Alembert paradox. Flow over a rotating cylinder is used to explain the Magnus effect and the

generation of lift. Compressibility effects have been studied by Janzen, Rayleigh, Imai and others [4]. The

critical Mach number was predicted accurately by van Dyke using computer series extension techniques [5].

Numerical solutions of Euler equations are reported inmany papers, see for example [6–9]. The simulation

of inviscid flows over a cylinder can be considered a benchmark problem to test new Euler codes. Moreover,any airfoil (or any closed curve) can be mapped to a circle. Nonuniqueness of transonic flows are also studied

based on the numerical solutions of potential and Euler equations for flows over a cylinder [10,11].

In the following, we present numerical solutions of potential (and stream function) equations with and

without entropy and vorticity corrections in a wide range of Mach numbers. The present hierarchical

formulation is based on Helmholtz decomposition of the velocity field into a gradient of a potential plus a

* Corresponding author. Tel.: +1-530-752-0212; fax: +1-530-752-4158.

E-mail address: [email protected] (M. Hafez).

0045-7825/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2003.12.048

1982 M. Hafez, E. Wahba / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1981–1995

vortical component. The continuity equation is solved for the potential function, the tangential momentumequation is solved for entropy, while the normal momentum equation provides the vorticity or the rota-

tional component of the velocity. The details of the governing equations and the boundary conditions are

given in the next section and the main numerical results are discussed afterwards.

2. Irrotational incompressible flows

For steady, two-dimensional, inviscid, adiabatic and incompressible flows with uniform upstreamconditions, the potential and stream functions are governed by Laplace equations. Analytical solutions can

be obtained via the superposition principle. Adding the solutions of a uniform flow and a doublet yields the

solution for an irrotational flow over a cylinder where the strength of the doublet is determined in terms of

the radius of the cylinder and the speed of the uniform flow. A potential vortex is needed to simulate the

effect of the cylinder rotation, where the strength of the vortex is related to the angular velocity of the

cylinder. Numerical solutions can be easily obtained if either the circulation or the angle of the stagnation

point is given (notice that the potential is a discontinuous function due to the circulation). For reasonable

grids, the numerical results agree well with the analytical solutions.

3. A cylinder in incompressible shear flows

If the incoming flow has vorticity, the potential function alone is not sufficient to represent such a flow.

For two-dimensional (and axisymmetric) flows, the stream function can still be used, and it is governed by a

Poisson�s equation where vorticity is the forcing function. Since the vorticity remains constant along a

streamline, the right-hand side of the Poisson�s equation depends on the stream function and this relationcan be found from the incoming flow conditions. The problem becomes nonlinear and can be easily solved

using iterative methods. For the special case of linear shear flow (constant vorticity) over half a cylinder, the

analytical solution is available [12]. The streamlines are plotted in Fig. 1. For the case of a full cylinder and

multiply connected domain problems, the reader is referred to [13].

-2 -1 0 1 20

0.5

1

1.5

2

2.5

3

3.5

4

X

Y

Fig. 1. Streamlines (incompressible flow).

1 2 3 4 5-0.01

0

0.01

0.02

0.03

0.04

0.05

X

u/U

∞

Stream FunctionOne Rotational Component

Fig. 2. Velocity at axis of wake (incompressible flow).

M. Hafez, E. Wahba / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1981–1995 1983

Alternatively, the solution can be obtained using a potential function with a correction to the velocity

due to the vorticity. In general, let

q ¼ r/þ q�; ð1Þwhere

x ¼ r� q�: ð2ÞThe potential is now governed by a Poisson�s equation:

r2/ ¼ �r � q� ð3Þand the boundary conditions on the solid surface becomes

q � n ¼ o/on

þ q� � n ¼ 0; ð4Þ

where n is the unit normal vector.

The two components of q� at the outer boundary are determined from the far field conditions where r/is assumed to present a uniform flow. Since the vorticity is a scalar for two-dimensional problems, one

component of q�, say, the h component, is obtained in the field by integrating Eq. (2) in the r-direction while

the other component remains unchanged as determined by the farfield conditions. The results of this

formulation is shown in Fig. 2 where the velocity at the axis of the wake is compared with the analyticalsolution for the special case of constant vorticity [12]. Figs. 3 and 4 shows the variation of the stagnation

point and the reattachment angle with vorticity. This simple example of inviscid separation, with closed

streamlines, is obviously independent of viscosity and the no slip condition. In general, the solution of the

steady Euler equations is not unique and depends on the location of the stagnation point on the cylinder

[14]. Moreover, the time dependent equations may not have a steady state solution.

4. Isentropic irrotational compressible flows

The effect of density variation will be studied next. For irrotational flows ðx ¼ 0Þ, the potential and

stream function equations are

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ωR/Uo

d sta

g

AnalyticalNumerical

Fig. 3. Stagnation point (incompressible flow).

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

45

50

ωR/Uo

θ (

deg)

AnalyticalNumerical

Fig. 4. Reattachment angle (incompressible flow).

1984 M. Hafez, E. Wahba / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1981–1995

r � qr/ ¼ 0; ð5Þ

r � rwq

� �¼ 0; ð6Þ

where the density is related to the speed via Bernoulli�s law assuming isentropic conditions:

q ¼ 1

�� c� 1

2M2

1ðq2 � 1Þ� 1

c�1

; ð7Þ

M. Hafez, E. Wahba / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1981–1995 1985

where the velocity is the gradient of a potential function

q ¼ r/: ð8ÞIn terms of a stream function, one can define a vector w with zero components in the plane and define the

flux as the curl of this vector

qq ¼ r� w; ð9Þwhere

w ¼ ð0; 0;wÞ: ð10ÞEqs. (5) and (6) are elliptic for purely subsonic flows and standard methods are applicable in a straight

forward manner for both cases.

For transonic flows, the equations are of mixed type; hyperbolic in the supersonic region and elliptic inthe subsonic one. For the potential equation, two necessary modifications are introduced for the simulation

of locally supersonic flow. The modified continuity equation reads

qt þr � qr/ ¼ r � �rq: ð11ÞThe right-hand side is an artificial viscosity term, while qt is an unsteady flow term to ensure a well posed

problem in an artificial time, representing the evolution of the solution through iterations. The artificial

viscosity has a cross wind component which can be eliminated and only ð�qsÞs is kept, where s is thestreamwise direction. The magnitude also can be adjusted to produce an upwind scheme similar to that of

Jameson [15], see also [16]. Notice Eq. (11) can be rewritten for smooth flows in the form:

1

a2ð/tt þ 2q/stÞ ¼ ð1�M2Þ/ss þ /nn: ð12Þ

The /tt is not essential. The two required modifications are the /st term and the artificial dissipationcorresponding to upwinding the term ð1�M2Þ/ss.

Alternatively, the artificial dissipation can be introduced through modifying the density [17] or the flux

[18].

One can also use a modified Bernoulli�s law which can be obtained by integrating the momentum

equation including artificial viscous terms or from the energy equation including viscous dissipation. In

either case, the density is related to the speed from the following relation:

/t þ1

2jr/j2 þ qc�1

ðc� 1ÞM21� 1

ðc� 1ÞM21� 1

2¼ �r2/: ð13Þ

Unlike Eq. (7), Eq. (13) has the time dependent term /t and the viscous term �r2/. In fact, Eqs. (11) and

(13) are used as a system of equations in q and / to calculate unsteady transonic flows [19]. However, in this

approach, artificial viscosity is needed everywhere in the field whether the flow is subsonic or supersonic

and an extra viscosity is needed in the supersonic flow region. Eliminating the density and solving one

equation for / is more advantageous, for both steady and unsteady flow calculations (see [20,21]).In this study, we shift the flux following [18], where the modified density is given by

�q ¼ q� ðqqÞsq

Ds ð14Þ

and

qq ¼ q�q� if M 6 1; ð15Þ

qq ¼ qq if M > 1: ð16Þ

1986 M. Hafez, E. Wahba / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1981–1995

In Eq. (15), q�q� is the flux at sonic conditions. With the modified density, standard discretization schemes

are applicable for the mixed type equation, the iterative scheme, on the other hand, must also be modified

explicitly or implicitly to introduce the /st term (instead of /t) in the calculations of supersonic flows.

For the stream function equation, another difficulty exists. The density is a double valued function of the

flux, with a square root singularity at the sonic condition. There is a maximum value for the flux (chocking)

and the density is imaginary beyond this maximum value. Following [22,23] we evaluate the velocity

components by integrating the equation for vorticity definition and then evaluate the density in terms of the

speed rather than the flux.The results for irrotational flows at M1 ¼ 0:5 are shown in Figs. 5 and 6. The present solutions are in

agreement with results available in literature.

-1 0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

3

θ (rad)

Mac

h N

umbe

r

Potential FunctionStream Function

Fig. 5. Mach distribution on the surface ðM1 ¼ 0:5Þ.

-4 -2 0 2 4

-4

-3

-2

-1

0

1

2

3

4

X

Y

Fig. 6. Mach contours (M1 ¼ 0:5, potential flow).

M. Hafez, E. Wahba / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1981–1995 1987

5. Nonisentropic rotational compressible flows

The conservation laws for mass, momentum and energy, written in vector notation, are

r � qq ¼ 0; ð17Þ

r � qrq ¼ �rp; ð18Þ

r � qqH ¼ 0: ð19ÞFor smooth 2-D flows, the tangential and normal momentum equations in terms of Crocco�s natural

coordinates s, n are given by

qqqs ¼ �ps; ð20Þ

qq2

R¼ �pn; ð21Þ

where R is the radius of curvature of the streamline.

The energy equation becomes

Hs ¼ 0: ð22Þ

Eqs. (20) and (21) are equivalent to

Ss ¼ 0; ð23Þ

xq ¼ ToSon

� oHon

; ð24Þ

where

x ¼ � oqon

þ qR: ð25Þ

In the above equations, x is the vorticity and S is the entropy. Therefore, the total enthalpy as well as the

entropy remain constant along streamlines and the vorticity is related to their gradients normal to the

streamlines. Across the shock, entropy jumps and the vorticity is generated if the shock is curved.In the present approach, the velocity is decomposed into the gradient of a potential plus a rotational

component as given by Eqs. (1) and (2). The continuity equation becomes

r � qr/ ¼ �r � qq�: ð26Þ

The density and the pressure are written in the form

p ¼ pie�DSR ; q ¼ qie

�DSR ; pi ¼

qci

cM21

ð27Þ

and

H ¼ cc� 1

piqi

þ 1

2q2: ð28Þ

The total enthalpy is obtained from the energy equation, the entropy is obtained from the tangential

momentum equation, while the normal momentum equation provides the vorticity. The rotational

1988 M. Hafez, E. Wahba / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1981–1995

component of the velocity is updated via integrating the equation for the vorticity inwards from the far field

to the body. To use the conservation forms, the deferred correction formulation is implemented, hence the

correction for the entropy is calculated based on the residual of the tangential momentum, including

artificial dissipation. Similarly, the correction to the vorticity is calculated based on the residual of the

normal momentum equation.

Since the tangential and normal momentum equations are combinations of the two components of the

vector momentum, the present formulation is completely equivalent to the solution of the conservation

laws. No special treatment is required, however, at stagnation points thanks to the introduction of artificialviscosity to the correction equations.

The present hierarchical formulation reduces to the potential equation if entropy and vorticity variations

vanish. The isentropic flow condition is recovered and the right-hand side of Eq. (26) disappears. One

option is to keep the entropy correction and to ignore the vorticity since it is higher order for many

applications as in [23,24]. (As an example, for a supersonic flow over a thin wedge and across the attached

shock, the entropy jumps but there is no vorticity since the shock is not curved.)

Standard numerical techniques are applicable to calculate the convection of entropy and to the inte-

gration of the vorticity definition to obtain the rotational velocity component. Transonic and supersonicflows over a cylinder in uniform flows are calculated as indicated above and the results are shown in Figs.

5–16.

In Fig. 7, a bubble of closed streamlines is shown for the transonic flow case ðM1 ¼ 0:5Þ. The time

dependent Euler codes do not have, however, a steady solution as indicated in [7,8]. For the supersonic flow

cases ðM1 ¼ 1:7; 2:0; 2:3Þ, the sonic lines are plotted in Fig. 13. The pressure at the axis for ðM1 ¼ 2:0Þ isshown in Fig. 14. The shock jump conditions and the standoff distance are plotted in Figs. 15 and 16. The

experimental data can found in [26]. The sonic point location on the cylinder depends slightly on the free

stream Mach number but this is not seen in the figure.Similarly, one can add the entropy and vorticity corrections to the stream function equation, see for

example [22]. In this case, there is no need to decompose the velocity into rotational and irrotational

components.

-2 -1 0 1 2-2.5

-2

-1.5

-1

-0.5

0

0.5

1.5

1

2

2.5

X

Y

Fig. 7. Streamlines (M1 ¼ 0:5, rotational flow).

-4 -2 0 2 4

-4

-3

-2

-1

0

1

2

3

4

X

Y

Fig. 8. Mach contours (M1 ¼ 0:5, rotational flow).

0 1 2 3 4 5 6-2

-1

0

1

2

3

4

5

6

7

θ, rad

p

Potential FlowRotational Flow

–C

Fig. 9. Cp on the surface ðM1 ¼ 0:5Þ.

M. Hafez, E. Wahba / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1981–1995 1989

6. A cylinder in compressible shear flow

In [25], hypersonic shear flows over a cylinder are simulated based on Euler and Navier–Stokes equa-

tions. The two solutions agree well for high Reynolds number flows in terms of standoff distance and the

size of the separation bubble. Here, we simulate transonic and supersonic flows over a cylinder based on the

present formulation. The results are presented in Figs. 17–20. The incoming flow for the transonic case has

a linear shear profile which varies over a unit height from 1 at uniform flow ðM1 ¼ 0:5Þ to 0.9 at the axis,

while for the supersonic case it varies from 1 at uniform flow ðM1 ¼ 1:7Þ to 0.5 at the axis. Hence, at theaxis, the flow is subsonic and the shock disappears. In our calculations, the total enthalpy is constant

-8 -6 -4 -2 0-8

-6

-4

-2

0

2

4

6

8

Fig. 10. Mach contours (M1 ¼ 2:0, potential flow).

-8 -6 -4 -2 0-8

-6

-4

-2

0

2

4

6

8

Fig. 11. Mach contours (M1 ¼ 2:0, rotational flow).

1990 M. Hafez, E. Wahba / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1981–1995

everywhere, while the entropy gradient in the incoming flow is related to the vorticity. On the other hand,the incoming shear flow may have constant entropy and variable total enthalpy.

7. Discussion of the results and concluding remarks

The results presented in this paper indicate that we can simulate inviscid flows across the Mach number

range, satisfying conservation laws of mass, momentum and energy by solving the potential equation to-

gether with entropy and vorticity corrections. The advantages of this hierarchical approach are obvious for

-8 -6 -4 -2 0-8

-6

-4

-2

0

2

4

6

8

Fig. 12. e�DS=R contours (M1 ¼ 2:0, rotational flow).

-8 -6 -4 -2 0-8

-6

-4

-2

0

2

4

6

8

Fig. 13. Sonic lines (rotational flow, M1 ¼ 1:7; 2:0; 2:3).

M. Hafez, E. Wahba / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1981–1995 1991

purely subsonic flows, where the potential equation is elliptic and standard numerical schemes and iterativealgorithms are used. For transonic flows, the potential equation is of mixed type and some well known

modifications in the locally supersonic flow regions are necessary for the numerical stability of the calcu-

lations and to capture shock waves. The entropy, generated behind curved shocks is convected along the

streamlines and the vorticity is calculated in terms of the entropy gradients across the streamlines.

In many aerodynamic applications, most of the flow is subsonic, governed by the isentropic potential

equation, where no artificial viscosity is needed while in solving Euler equations, artificial dissipation is

needed everywhere. For supersonic flows, shocks are weak in the far field, hence entropy and vorticity

variations can be ignored and the problem is reduced again to the solution of the isentropic potentialequation.

-6 -5 -4 -3 -20

1

2

3

4

5

6

7

8

9

10

X

P/P

∞

Rotational FlowPotential Flow

Fig. 14. Pressure at the axis ðM1 ¼ 2:0Þ.

1 1.5 2 2.5 31

2

3

4

5

6

7

8

9

10

11

P2/

P1

M1

AnalyticalNumerical

Fig. 15. Pressure ratio at axis.

1 2 3 4 5 6 7 80.05

0.0750.1

0.25

0.50.75

1

2.5

57.510

M∞

δ/d

ExperimentalRotational FlowPotential Flow

Fig. 16. Standoff distance.

1992 M. Hafez, E. Wahba / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1981–1995

-2 -1 0 1 2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

X

Y

Fig. 17. Streamlines (transonic shear flow).

-4 -2 0 2 4

-4

-3

-2

-1

0

1

2

3

4

Y

Fig. 18. Mach contours (transonic shear flow).

M. Hafez, E. Wahba / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1981–1995 1993

We used the present formulation to simulate uniform and shear flows over a cylinder. All the reported

results converged to a maximum residual of 10�8 using a finite volume discretization and a line relaxation

procedure. There are some interesting features in these calculations. The inviscid separation, the formationof supersonic bubbles and the geometry of the sonic and shock curves and their intersections. Some

comparisons with available data in the literature are shown to confirm our results. The assumption of a

steady flow is a limiting factor however, and the extension to unsteady flow formulation is required,

particularly for the study of vortex shedding and the inviscid analog of Karman street. The extension to

viscous flow simulation is the subject of a separate paper [27].

-8 -6 -4 -2 0

-8

-6

-4

-2

0

2

4

6

8

Fig. 20. Mach contours (supersonic shear flow).

-2 -1 0-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Fig. 19. Streamlines (supersonic shear flow).

1994 M. Hafez, E. Wahba / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1981–1995

References

[1] M.M. Zdravkovich (Ed.), Flow around Circular Cylinders, vols. I and II, Oxford University Press, 2003.

[2] M. van Dyke, An Album of Fluid Motion, Parabolic Press, Stanford, 1982.

[3] O. Rodriguez, The circular cylinder in subsonic and transonic flow, AIAA J. 22 (1984) 1713–1718.

[4] I. Imai, Approximation Methods in Compressible Fluid Dynamics, University of Maryland, Institute for Fluid Dynamics and

Applied Mathematics, 1957.

[5] M. van Dyke, Perturbation Methods in Fluid Dynamics, Parabolic Press, Stanford, 1975.

[6] T.J.R. Hughes, T.E. Tezduyar, Finite element methods for first-order hyperbolic systems with particular emphasis on the

compressible Euler equations, Comput. Methods Appl. Mech. Engrg. 45 (1984) 217–284.

[7] N. Botta, The inviscid transonic flow about a cylinder, J. Fluid Mech. 301 (1995) 225–250.

M. Hafez, E. Wahba / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1981–1995 1995

[8] M. Pandolfi, F. Larocca, Transonic flow about a circular cylinder, Comput. Fluids 17 (1989) 205–220.

[9] M.D. Salas, Recent developments in transonic Euler flow over a circular cylinder, Math. Comput. Simulat. (25) (1983).

[10] M.M. Hafez, A. Dimanlig, Some anomalies of the numerical solution of the Euler equations, Acta Mech. 119 (1996) 231–239.

[11] M.M. Hafez, A. Dimanlig, Simulations of compressible inviscid flows over stationary and rotating cylinders, Acta Mech. (1994)

241–249.

[12] L.E. Fraenkel, On corner eddies in plane inviscid shear flow, J. Fluid Mech. 11 (1961) 400–406.

[13] T.E. Tezduyar, Finite element formulation for the vorticity-stream function form of the incompressible Euler equations on

multiply-connected domains, Comput. Methods Appl. Mech. Engrg. 73 (1989) 331–339.

[14] C.H. Bruneau, J.J. Chattot, J. Laminie, R. Temam, Numerical solutions of Euler equations with separation by a finite element

method, in: Proceedings of the Ninth International Conference on Numerical Methods in Fluid Dynamics, Lecture Notes in

Physics, vol. 218, Springer Verlag, 1984.

[15] A. Jameson, Iterative solutions of transonic flows over airfoils and wings including flows at mach1, Commun. Pure Appl. Math. 27

(1974) 283–309.

[16] C. Hirsch (Ed.), Numerical Computation of Internal and External Flows, vols. 1 and 2, 1990.

[17] M.M. Hafez, J. South, E. Murman, Artificial compressibility methods for the numerical solution of the full potential equation,

AIAA J. 17 (1979) 838–844.

[18] M.M. Hafez, W. Whitlow, S. Osher, Improved finite difference schemes for transonic potential calculations, AIAA J. 25 (11)

(1987) 1456–1462.

[19] R. Chipman, A. Jameson, Alternating-direction implicit algorithm for unsteady potential flow, AIAA J. 20 (1982) 18–24.

[20] V. Shankar, I. Hiroshi, J. Gorski, S. Osher, A fast time accurate unsteady full potential scheme, AIAA Paper, 85-1512.

[21] W. Whitlow, M.M. Hafez, S. Osher, An entropy correction method for unsteady full potential flows with strong shocks, NASA

TM 87767, 1986.

[22] M.M. Hafez, D. Lovell, Numerical solution of transonic stream function equation, AIAA J. 21 (1983) 327–335.

[23] M.M. Hafez, D. Lovell, Entropy and vorticity corrections for transonic flows, Int. J. Num. Methods Fluids 8 (1988) 31–53.

[24] M.M. Hafez, D. Lovell, Transonic small disturbance calculations including entropy corrections, in: Num. and Phys. Aspects of

Aero Flows Conf., Springer, 1981.

[25] A. Kumar, M.D. Salas, Euler and Navier–Stokes solutions for supersonic shear flow past a circular cylinder, AIAA J. 23 (1985)

583–587.

[26] H. Liepmann, A. Roshko, Elements of Gas Dynamics, 1957.

[27] M. Hafez, E. Wahba, Hierarchical formulations for transonic flow simulations, CFD J. (2003) 377–382.