an improved unsteady, unstructured artificial compressible finite volume scheme for viscous...

5/25/2018 An Improved Unsteady, Unstructured Artificial Compressible Finite Volume Scheme for Viscous Incompressible Flows

1/20

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2002; 54:695714 (DOI: 10.1002/nme.447)

An improved unsteady, unstructured, articial compressibility,

nite volume scheme for viscous incompressibleows: Part I. Theory and implementation

A. G. Malan1, R. W. Lewis1;; and P. Nithiarasu2

1Department of Mechanical Engineering; University of Wales Swansea; Singleton Park;Swansea SA2 8PP; U.K.

2Department of Civil Engineering; University of Wales Swansea; Singleton Park; Swansea SA2 8PP; U.K.

SUMMARYA robust, articial compressibility scheme has been developed for modelling laminar steady stateand transient, incompressible ows over a wide range of Reynolds and Rayleigh numbers. Articialcompressibility is applied in a consistant manner resulting in a system of preconditioned governingequations. A locally generalized preconditioner is introduced, designed to be robust and oer goodconvergence rates. Free articial compressibility parameters in the equations are automated to allowease of use while facilitating improved or comparable convergence rates as compared with the standardarticial compressibility scheme. Memory eciency is achieved through a multistage, pseudo-time-explicit time-marching solution procedure. A node-centred dual-cell edge-based nite volume discretiza-tion technique, suitable for unstructured grids, is used due to its computational eciency andhigh-resolution spatial accuracy. In the interest of computational eciency and ease of implementa-tion, stabilization is achieved via a scalar-valued articial dissipation scheme. Temporal accuracy isfacilitated by employing a second-order accurate, dual-time-stepping method. In this part of the paperthe theory and implementation details are discussed. In Part II, the scheme will be applied to a number

of example problems to solve ows over a wide range of Reynolds and Rayleigh numbers. Copyright? 2002 John Wiley & Sons, Ltd.

KEY WORDS: incompressible ow; unstructured; nite volume; articial compressibility; preconditioning

1. INTRODUCTION

Numerical simulation of incompressible uid ow is of great practical importance due to itsmany industrial applications. These range from hydrodynamics and low-speed aerodynamics

to the natural and forced convection systems found in heat exchangers and electronic coolingdevices. The involved spectrum of ow regimes and wide range of length scales has resulted

Correspondence to: R. W. Lewis, Department of Mechanical Engineering, University of Wales Swansea, SingletonPark, Swansea SA2 8PP, U.K.

E-mail: [email protected]

Received 12 June 2001Copyright ? 2002 John Wiley & Sons, Ltd. Revised 4 September 2001


2/20

696 A. G. MALAN, R. W. LEWIS AND P. NITHIARASU

in the need for a general numerical tool which may be readily applied to simulate theextensive range of ow conditions without prior ad hoc modications. Such a tool shouldfurther oer spatial and temporal accuracy, eect memory eciency and be suitable formassive parallelization typically required for industrial three-dimensional viscous uid ow

problems.The two main approaches used in solving incompressible ow problems are the so-called

pressure-based (projection scheme) proposed by Patankar [1] and density-based (articialcompressibility) method introduced by Chorin [2]. Historically, the former was initially themost extensively used while articial compressibility (AC) has only received comparableresearch attention in recent years. This is in part due to the success achieved with transonicow time-marching schemes, which aided in bringing to the front the main advantages of theAC method, viz: (a) memory eciency and suitability for application to massively parallelenvironments and (b) mutual technology transfer with compressible time-marching systems.

The concept of Chorins [2] AC scheme hinges on the introduction of an articial relationbetween pressure and density through the addition of a pseudo-time pressure derivative termto the continuity equation. A number of recent publications, viz. Belov et al. [3], Manzari [4],Drikakis et al. [5] and Zhao and Zhang [6], demonstrate the application of the scheme inthis form (hereafter referred to as the standard scheme). The standard scheme, however,exhibits a number of deciencies. Tamamidis et al. [7] and Ferziger and Peric [8] have, forexample, indicated that the choice of the AC parameter has a signicant eect on stabilityand convergence, while its choice requires considerable experience and intuition on behalfof the user [9]. Further, most of the methods found in the literature are applied to solveincompressible ows only over a narrow band of Reynolds numbers, being either mediumrange or highly convective. Schemes applied to low Reynolds number ows often suereither from a deterioration in accuracy [4] or slow convergence rates [6].

As mentioned above, the choice of the free AC parameter (pseudo-acoustic velocity) playsan important role in the convergence and stability of the scheme. To avoid the need for

numerical experimentation and ensure general applicability, this parameter needs to beautomated at each computational cell in the domain. Consideration must be given to localconvective velocities [10] as well as diusion velocities as pointed out by Choi and Merkle[11]. The way in which these aspects are addressed should be appropriate to the preconditioning

parameter used and, in our experience in the case of pseudo-explicit schemes, the type ofstabilization used.

The standard scheme eectively only treats the transient term in the continuity equation.An alternative is to apply the concept of AC to all transient terms in the governingequations in a more consistent manner resulting in a local preconditioned system. The latterwas rst documented by Turkel [10] and shown to exhibit dierent numerical characteris-tics to that of the standard method. Weiss and Smith [12] have, for example, demonstrateda signicant enhancement in convergence rates in certain cases. Turkel also introduced a

preconditioning parameter which results in isotropic error waves, further improving conver-gence. A drawback of this preconditioning parameter is, however, that it results in a less robustsystem as compared with the standard scheme as documented by Turkel [13] and Michelassiet al. [14].

On the subject of discretization, the pseudo-temporal component should be of a form toexploit the advantages of the AC scheme. This is facilitated through the use of an explicit

Copyright ? 2002 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng2002; 54:695714


3/20

VISCOUS INCOMPRESSIBLE FLOWS: PART I. THEORY AND IMPLEMENTATION 697

pseudo-time-marching method. Given an apt choice for the AC parameter, the resulting systemof equations can readily be solved in pseudo-time [2; 15]. Spatial discretization should beapplicable to unstructured and solution-adapted grids. Lewis et al. [16] and Nithiarasu andZienkiewicz [17] have indicated the latter to be of importance and show adapted grids to

be potentially both accurate and computationally ecient. The vertex-centred dual-cell edge-basednite volume scheme is a relatively recently developed unstructured spatial discretizationmethod which oers accuracy, computational eciency and requires less memory than cell-

based counterparts on unstructured grids [18]. The method has to date not been applied to anAC pseudo-explicit scheme.

The addition of stabilization terms is necessitated by the collective nature of the spa-tial and pseudo-temporal discretization described above. Lin and Sotiropoulos [19] inves-tigated the suitability of a number of articial dissipation techniques for this purpose. Ofthese, the scalar dissipation model introduced by Jameson et al. [20] oers a balance be-tween accuracy, simplicity and computational eciency. Mavriplis [21] proposed a methodfor its application to unstructured vertex-centred grids for compressible ow. This methodis yet to be applied to incompressible ow on a dual-cell vertex-based discretizationscheme.

For transient problems, many researchers employ a dual-time-stepping technique as usedby Shuen et al. [22]. This involves introducing a real-time outer loop to the above explicitpseudo-time-stepping system in an implicit manner. The addition of the outer loop renderspseudo-time temporal accuracy of no importance, and AC is implemented together with con-vergence enhancement techniques such as local time-stepping, without detriment to real-timetemporal accuracy. Owing to its implicit nature, the scheme does not suer from the real-time-step limitations of explicit counterparts, resulting in a signicant saving in computationaleort. Second-order temporal accuracy is further facilitated through the use of a second-order

backward dierence operator.In summary, it is clear from the available literature that current pseudo-explicit AC schemes

have diculty in solving a wide range of incompressible ows. The vast majority are furthernot straightforward to use and often require signicant insight and experience from the user.Renements such as local preconditioning, may deteriorate robustness. No one scheme hasfurther, to the authors knowledge, been applied with success to a wide range of Rayleighand Reynolds numbers, ranging from purely viscous Stokes ow to high Reynolds numberconvective ows. In this paper, we attempt to address these issues through the developmentof an improved unsteady AC scheme.

First, a locally generalized preconditioner is introduced to eect enhanced convergence whilemaintaining robustness. Second, it is endeavoured to automate free AC parameters in orderto facilitate optimal or near-optimal convergence as compared with the manually optimizedstandard scheme. Third, special treatment is given to the construction of articial dissipationin the interest of consistency with preconditioning. Spatial discretization is performed using

a node-centred dual-cell edge-based nite volume scheme and a scalar-valued articial dissi-pation operator proposed for compressible ow cell-centred schemes extended for use on thevertex-based scheme.

In Part II of the paper, the developed scheme is applied to a number of benchmark problemsto solve ow over a wide range of Reynolds and Rayleigh numbers. In all cases convergencecomparisons are made with the standard AC scheme.



4/20


2. GOVERNING EQUATIONS

The NavierStokes equation describing the non-isothermal laminar ow of a uid consists ofequations enforcing the principles of mass, momentum and energy conservation. Assuming a

Newtonian uid with negligible thermal viscous dissipation, the governing system of equationsmay be written for a Cartesian co-ordinate system in non-dimensional conservative form as

@W

@t +

@Fj

@xj @G

j

@xj=S (1)

where

W=

u1

u2

u3

T

; Fj=

uj

u1uj

u2uj

u3uj

Tuj

; Gj=

0

1j

2j

3j

1

Re Pr

@T

@xj

; S=

0

0

0

GrRe

T

0

(2)

and is the density; uj the velocity component in direction xj; p the pressure and T thetemperature. Non-dimensional quantities are related to their dimensional counterparts (depictedwith superscript ) through the following relations:

t= tV0 =L; uj= u

j=V

0 ; p = p=0 V

20 ; T= (T

T0 )=(TW T0 ) = =0 ; xj=x

j =L; = =0 (3)

In Equation (1), S depicts a gravitational buoyancy force in direction x2. Here, the densityvariation is interpolated by invoking the Boussinesq approximation. The Reynolds, Prandtland Grashof numbers are dened as

Re =0 V

0 L

0; Pr=

0C

p

k; Gr=

L320 g(TT)

20(4)

where V0 and L are respectively, the characteristic velocity and length used for non-

dimensionalization. 0 ;

0 and k are the density, molecular dynamic viscosity and ther-

mal conductivity, respectively. The reference temperature is given by T0 and C

p depicts thespecic heat. T and g are the coecients of thermal expansion and gravitational acceleration,respectively.

The stress term ij is dened as

ij=

Re

@ui@xj

+@uj@xi

23

@uk@xk

ij

pij (5)

where ij is the Kronecker delta and is equal to unity when i =j and equal to zero wheni=j.



5/20


3. IMPLEMENTATION OF ARTIFICIAL COMPRESSIBILITY

3.1. Introduction of articial compressibility

Equation (1) is numerically problematic when applied to incompressible ows. This is due tothe large disparity between the acoustic and convective velocities resulting in an extremelysti system. Chorin [2] proposed a method for addressing this through the introduction of anarticial relation between density and pressure and christened the method articial compress-ibility. The proposed method involved replacing the density in the time derivative term by(1=2)p where (1=2) is the articial compressibility. Having altered the acoustic velocity,the time accuracy of the equation is destroyed and time t is replaced with pseudo-time t.

In summary, AC in essence comprises altering the relation between density and pressure inpseudo-time, viz. = f(p (t)). The concept may be implemented into the continuity equationthrough the use of a dierential operator as follows:

@

@t =

@

@p

@p

@t

Now, by applying the above dierential operator consistently to all transient terms inEquation (1) similar to Weiss and Smith [12], and replacing t with t we may write

@W

@Q

@Q

@t+

@Fj

@xj @G

j

@xj=S (6)

where Q= (p; u1; u2; u3; T)T and

@W

@Q=

@

@p

0 0 0 0

@

@pu1 0 0 0

@

@pu2 0 0 0

@

@pu3 0 0 0

@

@pT 0 0 0

(7)

The dependent variable Q is now similar to that solved for in the standard articial com-pressibility approach and hold advantages such as elimination of the possibility for timereversals in the viscous terms, as pointed out by Choi and Merkle [11].

To complete the introduction of AC, the term@=@p, which is equal to 1=c2 for compressibleows, is replaced by the pseudo-acoustic velocity counterpart, viz. 1=c2 . c (equivalent to used above) must be chosen for each problem and is also referred to as the free AC parameter.



6/20


3.2. Generalized preconditioning

Turkel [10] proposed a technique to generalize the preconditioning matrix by introducingan arbitrary constant which facilitates switching between dierent types of preconditioners.

Introducing c into Equation (7), and generalizing via the addition of two arbitrary constantsau (proposed by Turkel) and aT we write

= @W

@Q

=

1=c2 0 0 0 0

auu1=c2 0 0 0

auu2=c2 0 0 0

auu3=c2 0 0 0

aTT=c2 0 0 0

(8)

where

aT=

0 if au= 0

1 if au0

By setting au= 0, the standard scheme is recovered while au= 1 corresponds to the formgiven above (Equation (7)). Turkel [10] proposes setting au= 2, which results in improvedconvergence rates due to the condition number approaching unity. For this reason, others suchas Michelassi et al. [14] and Liu et al. [23] have used the preconditioner in this form. Turkel[13] has, however, shown that setting au = 2, does deteriorate the robustness of the schemeas compared with au = 0. In our experience (through numerical experimentation) this is alsoless robust than setting a

u= 1. The preconditioner resulting from the three values for a

u is

summarized as follows:

au = 0: Standard scheme. The scheme is more robust as compared with au =2 but mayresult in poorer convergence rates.

au = 1: Consistently introduced AC scheme. Turkels preconditioner (au = 2): Results in improved convergence rates as compared

with the above. Through numerical experimentation it is, however, found to be the leastrobust of the three.

The reason for the less robust nature of the preconditioner resulting in faster convergenceis not well understood, and, therefore, dicult to address. In the next section, we will forwardour view on this and propose a method for calculating au locally at each pseudo-time step.

We will refer to this method as locally generalized preconditioning (LGP).

3.3. Locally generalized preconditioner (LGP)

In developing the LGP, we rst attempt to gain insight into why certain values for thepreconditioning parameter au are more prone to instability in pseudo-time. To evaluate theeect that the preconditioner has on the change in primitive variables at each pseudo-time



7/20


step, it is instructive to look at the inverted form of Equation (8)

1 =

c2 0 0 0 0

auu1= 1= 0 0 0auu2= 0 1= 0 0

auu3= 0 0 1= 0

aTT= 0 0 0 1=

(9)

What is evident is that au serves to add spatial terms of the continuity equation to theother (momentum and energy) equations. Oscillations in the continuity equation will thushave a direct eect on the other equations possibly eecting stability. For au =0 this eectis cancelled while when au = 2 it is most pronounced. This is consistent with the observationmentioned above.

As before we prefer large au values (as close as possible to au =2) in the interest ofconvergence. This may, however, detrimentally eect the overall scheme stability, especiallywhen the computational domain contains regions with high primitive variable gradients. It is,therefore, advisable to devise a method of establishing au values locally.

It is proposed to utilize a pressure switch to gauge evolving gradients in pseudo-time ateach point in the domain. The logic behind this being that this would oer an indication of

both pressure gradients as well as local oscillation in the continuity equation in pseudo-time.It is proposed that a pressure sensor, similar to the one used for transonic purposes [24],

be applied and the free preconditioning parameter au scaled accordingly.Dening the pressure sensor we write in one-dimensional space, where p(x) is the pressure

at x =xm (point m in space)

Apm = limxx m

+|p(x

m

) p(x)||p(xm)| |p(x)| (10)

where

p (xm)= limxx m

p(x) p(xm)x xm (11)

The preconditioning parameterau is now set to au =2(1Ap) which will vary from au = 2in smooth regions to au 0 in regions containing large gradients. Implementing this intoEquation (8) we write

LGP=@W

@Q

LGP

=

1=c2 0 0 0 0

2(1 Ap)u1=c2 0 0 02(1 Ap)u2=c2 0 0 0

2(1 Ap)u3=c2 0 0 0

aTT=c2 0 0 0

(12)



8/20


The choice of the pseudo-acoustic velocity is the remaining AC-related issue and is addressedin the next subsection.

3.4. Automation of pseudo-acoustic velocities and time step calculationsAs mentioned before, it is evident from the literature that the choice of the pseudo-acousticvelocity is of importance as it eects the stability and convergence characteristics of thescheme. In the interest of general applicability and ease of use, this parameter should ideally

be automated. A local treatment at each computational cell in the grid will be eected toensure applicability to problems containing ows with both diusion and convection dominatedregions.

The rst consideration is that the Mach number be kept below unity [2]. Next, incalculating c, a distinction is made between convective-dominated (high Reynolds numbers)and diusion-dominated (low Reynolds number) ow regions. This is as the pseudo-acousticvelocity c will be scaled in a dierent manner in each case. For the purpose of discussion,we designate the pseudo-acoustic velocity scaled in accordance with convective terms as cconv ,

and those scaled in accordance with diusion velocities by cdi.

3.4.1. Convection dominated ow. Considering rst high Reynolds number ows, Turkel [10]has shown that, in the interest of convergence rate, the parameter cconv , in the inviscid limit,should be kept as close as possible to the local convective velocity. This is implemented asfollows:

cconv=

conv if |u|6cconv|u| if |u|cconv

(13)

where|u|=ujuj andcconv is typically set to 105 Vmax with Vmax being the maximum velocitymagnitude in the eld. cconv is dened in this manner to ensure that the pseudo-acousticvelocities do not go to zero at stagnation points.

In the case of articial dissipation stabilized schemes, it is found by numerical experimen-tation that very small values forcconv , however, result in instabilities in pseudo-time, eectingconvergence. As a rst attempt at addressing this issue, we postulate that the instabilitiesare due to cconv being used to scale the dissipation terms (the stabilization scheme will bediscussed in a later section). This, together with a rst attempt at addressing this problem,follows.

In the case of low Reynolds number ows, the large local diusion velocities ensureadequately large values for the pseudo-acoustic (c) velocity (to be discussed below) andabove-mentioned articial dissipation scaling problem is of no concern. At intermediate andhigh Reynolds numbers this may not be the case. It may, therefore, be insightful to investigate

the relation between diusion velocities and Reynolds number by considering a hypotheticalcase of laminar ow over a at plate of unit length at zero incidence.

Of importance in the at plate problem are the diusion velocities within the boundary layerand how these reduce as the Reynolds number increases. This region is of importance as it,as mentioned before, contains large velocity gradients thus requiring smoothing. Following,we attempt to estimate the local diusion velocities within the boundary layer and how thesechange with the Reynolds number.



9/20


0

0.2

0.4

0.6

0.8

1

1 10 100 1000 10000

KinematicViscousVelocity

Log(Re)

x=0.1x=0.5x=0.8

Figure 1. Kinetic viscous velocity vs Reynolds number at distance x from leading edge.

Using the Blasius [25] solution, the 99 per cent boundary layer thickness, in non-dimensionalform as a function of Reynolds number and non-dimensional distance x1 from the leading edge,is calculated as follows:

99 per cent= 3:5

2x1Re

(14)

Using this, the kinetic viscous diusion velocity in the boundary layer may be calculated.

For the purpose of the calculation, we assume a grid with vertical edges orthogonal (in direc-tion x2) to the plate with six equally spaced nodes through the cross-section of the boundarylayer 99 per cent= 6x2. The diusion velocity is now given by

udi x1Re

(15)

where assuming that x1x2,

= x1x2x1+ x2

x2

By setting 99 per cent= 6x2 in Equation (14), rearranging and setting into Equation (15)

in terms of x2, we compute the diusion velocity as a function of the position from theleading edge of the plate and the Reynolds number as follows:

udi 1:212x1Re

(16)

The variation of udi with the Reynolds number is shown graphically in Figure 1. udiclearly drops drastically with an increase in the Reynolds number, and at Re =1000 is up to



10/20


Figure 2. Convergence history for lid-driven cavity, Re=1000 for various values for cconv .

an order of magnitude smaller than the free stream velocity. As the convective velocities aresimilarly small close to the plate, cconv is to be based on another quantity to avoid excessivedown-scaling of the articial dissipation in boundary layers and stagnation points. This may

be done through setting a minimum value i.e. a suitable value for cconv .Values used by other researchers for

cconv range from 0.3 [26] to 1.0 [12]. Although

robust, the latter value may result in large disparities between pseudo-acoustic and convectivevelocities for medium range Reynolds numbers (O(1000)), thereby slowing down the conver-gence rate. This is demonstrated in Figure 2 (L2 is the average norm of the dierence inthe pressure variable between successive iterations, normalized to that of the rst iteration),which shows the convergence for a lid-driven cavity problem (to be discussed in Part II) fordierent values of cconv .

Setting a xed minimum value forcconv that is notably smaller than the free stream velocitydoes not exhibit the above deciency but, however, may not be in the best interest of stabilityat all Reynolds numbers. In Figure 3 it is seen that for a value of 0.3 convergence is notachieved for the Re = 5000 case. To ensure convergence, we propose as a rst attempt tovary cconv with the Reynolds number as follows:

cconv =

Re=10000 Vmax Re61104

1:0 Vmax Re1104(17)

where the Reynolds number is representative of similar boundary layer length scales comparedto the at plate example. Note that in contrast to the work of Rizzi and Eriksson [26] theminimum value for c2conv extends above 0.3 to 1.0 at high Reynolds numbers. The latter is



11/20


Figure 3. Convergence history for lid-driven cavity, Re=5000 for various values for cconv .

consistent at high Reynolds numbers with Turkel [12] who proposed using a value of one atRe =500000.

3.4.2. Diusion-dominated ow. Low Reynolds number ow is generally diusion dominated.

As shown in Figure 1, the kinetic viscous diusion velocities typically increase with decreasein Reynolds number to a value above the free stream velocity in regions where the Reynoldsnumbers are of the order 10. The diusive and convective velocities in the boundary layerwill thus dier by orders of magnitude in certain regions. Pseudo-acoustic velocities computedfrom Equation (13), being far smaller than the local diusive velocities, will, as pointed out

by Choi and Merkle [11], signicantly impair the rate of convergence. This is dealt with byensuring that the maximum convective and diusion time steps are of similar magnitude.

The convective time step is, as is typical, dened as

t=CFL

max(18)

CFL is the CourantFriedrichsLewy number and the length scale is given by

=

1

xjxj

1

(19)

which is similar to that used in nite dierence electromagnetic wave propagation problems[27] and analogous to the eective element size used in explicit nite element computational



12/20


uid dynamics calculations [28]. The parametermax designates the maximum eigenvalue andis given by

max=1

2|u|(2 au) + |u|2(2 au)2 + 4c2 (20)

Choi and Merkle [11] dene the diusion time step tdi as a function of the grid size anddiusion velocity udi. We, however, prefer, consistent with the Eulerian approach, to denetdias a function of the diusion velocity in addition as follows:

tdi=

udivN

+ |u|1

(21)

where vN is the von Neumann number and in general taken as 0:5 [28]. udi is calculated as

udi=

Re (22)

For diusion dominated ows the pseudo-acoustic velocity is now computed by settingt equal to tdi (ensuring that these are of similar magnitudes) resulting in the followingexpression:

c2di =1

4

2CFL

tdi |u|(2 au)

2 |u|2(2 au)2 (23)

This concludes the calculation of cdi. In the general case, the pseudo-acoustic velocity iscalculated as c= max{cconv ; cdi}. Above relations are then applied to automate the calculationof c at each computational cell for a wide spectrum of Reynolds and Rayleigh numbers.

For the standard scheme (au= 0), the value of c giving optimal convergence is typi-cally obtained through numerical experimentation techniques. Two widely used methods are;

(a) using a constant value for c across the eld where c is found through numerical exper-imentation as used in recent publications by Tamamidis et al. [7] and Zhao and Zhang [6],and (b) using the relation c2 =max{0:3; rujuj} with 1r5 being obtained for each problemvia numerical experimentation as proposed by Rizzi and Eriksson [26]. These methodswill be employed in Part II for comparison with the proposed automation procedurein an attempt to evaluate its eectiveness in consistently facilitating comparable or improvedconvergence rates.

4. SOLUTION PROCEDURE

4.1. Spatial discretization

The rst stage in solving the system of governing equations is transformation into an integralor weak form. Equation (6) is integrated on an arbitrary control volume as follows:

LGP@Q

@td +

@Fj

@xj @G

j

@xj

d =

Sd (24)

where the state, ux and source vectors are dened in Equations (2)(5).



13/20


m

n

m

Figure 4. Vertex-centred dual cell nite volume (control volume) surrounding node m.

The nite volume approach involves casting the volume integrals containing spatial deriva-tives of the inviscid and viscous ux vectors, into a surface integral through the application ofthe divergence theorem. For an arbitrary volume , bounded by the surface , Equation (24)

now becomes

LGP@Q

@td +

(Fj Gj)nj d =

Sd (25)

where n is the unit vector in the direction normal to the boundary segment d.The spatial domain is subdivided into triangular cells (in 2D) and integrals applied to each

cell, as shown in Figure 4. The state variable vectorQ is stored at each node or vertex, and thesurface integrals computed in an edge-wise manner on the vertex-centred cells (volumes). Thevertex-centred dual cells are dened by connecting the mid-points of the edges and centroidsof the associated triangles. This technique facilitates edge-based storage which is ecient bothcomputationally and in terms of memory resources as compared with cell-based schemes.

Moving on to the computation of each term in Equation (25), the surface integrals are

rewritten in discrete form for vertex m as follows:m

nj dedges

mnCjmn+B

jm (26)

where

mn=12

(m+n)

and Cjmn= nj d is the edge coecient of the edge dened by nodes m-n. Summation isperformed over all edges connected to node m. Bm denotes the boundary contribution in thecase where m designates a boundary node and is computed as follows:

Bjm() =

bound: edges

14 (3m+n)

nj d

Volume integrals are computed by assuming that the nodal value represents an averageover the volume as follows:

m

d = m m mm (27)



14/20


Equation (27) is formally second-order accurate on a nite dierence-type grid. This conceptis also implemented in calculating the derivatives in the viscous stress terms (Equation (5))as follows:

@@xj

m

@@xj

m

= 1m

m

@@xj

d = 1m

m

nj d (28)

Above surface integral is now again discretized using Equation (26).The pressure sensor (Equation (10)) is calculated using a harmonic operator [24]:

Apm =|edges(pnpm)|

edges |pnpm| (29)

Implementing the above, Equation (24) discretized at node m now becomes

LGPm

dQm

dtm+

edges

[Fj

mn Gj

mn

]Cj

mn

+Bj

m

m

= [S]m (30)

where Bjm is non-zero only at boundary nodes.

4.2. Articial dissipation

Owing to the central dierence explicit nature of the AC scheme, stabilization is required toeliminate spurious oscillations. Scalar-valued dissipation, based on work done by Jameson [20]is implemented for this purpose and involves the addition of an articial dissipation term tothe right-hand side of the discretized governing equation (Equation (30)). The dissipation termD may be constructed on an unstructured vertex-centred grid through the use of a biharmonicoperator as proposed by Mavriplis [21]. This term is constructed on the vertex-based scheme

described above by forming an approximation to a double Laplacian (one inside another) onthe state variables at each node. The rst (inner) operator is calculated using an edge-basedapproach as follows:

2m=edges

(nm) (31)

where summation is performed over all edges connected to node m and typically designatesthe dependent variable vector. The biharmonic operator is constructed by essentially repeatingthe above step

Dm=4 m+ n2 edges

(2n 2m) (32)

where 4 is an empirical constant, the value of which is to be determined for each problemthrough numerical experimentation (the objective is to use the smallest possible value tostabilize and eliminate spurious oscillations). m is the isotropic scaling coecient at node mand is calculated as follows:

m=edges

|u Cmn| +c|Cmn| (33)



15/20


The current method used by many researchers to calculate the dissipation for application tothe standard AC scheme [4; 19; 29; 30] is by setting =Q. D is then added to the right-handside of Equation (30) after all terms have been premultiplied by 1m such that the transientterm reduces to (dQ=dt).

For the preconditioned system, a more consistent approach is proposed in this paper byadding the dissipation term to the right-hand side of Equation (1) and setting =W. Theuse of wj as a dependent variable is, however, not meaningful as the pressure dissipationterm will be zero for incompressible ows. The problem of the vanishing dissipation term issolved by the application of the AC concept to the above operator.

Considering a one-dimensional case, the Laplacian operator may be written as follows:

limx0

2Wx2

=

@2W

@x2

=

@2

@x2@2(u)

@x2@2(T)

@x2

T(34)

By now introducing AC in a spatial sense, Equation (34) becomes

@2W

@x2

=

@

@x

@

@p

@p

@x u

@

@p

@p

@x +

@u

@x T

@

@p

@p

@x+

@T

@x

T

= @

@x

1

c2

@p

@x

u

c2

@p

@x +

@u

@x

T

c2

@p

@x+

@T

@x

T

= limx0

1

x2Q

(35)

Writing this in discrete form, extending to multidimensions and implementing into the

approximate Laplacian operator, the rst (inner) approximate operator becomes

2Wm=edges

m(QnQm) (36)

The biharmonic operator (Equation (32)) is now constructed by setting2m=2Wm. Aspreviously, Dm is added to the right-hand side of Equation (30). Calculating the dissipationterm in this way has not, to the authors knowledge, to date been applied in this manner to anAC scheme. It proved signicant in the solution of pure viscous ows (Stokes ow) whichwill be demonstrated in Part II.

We digress briey to the point raised previously on the sensitivity of the stability of thescheme to small values of c. It is clear from Equation (32) that the dissipation term D will

be signicantly down-scaled in regions where both the local convective and pseudo-acousticvelocities are small. This will typically be the case inside boundary layers near the boundarywalls at high Reynolds numbers. The problem is that a small scaling factor () would workagainst the dissipation operators task of increasing D in regions where higher gradients are

present, resulting in possible instability. As this counter productive behaviour is expected toincrease with increasing Reynolds number, it was proposed to scale the minimum allowablevalue for c as a function of Reynolds number (Equation (17)).



16/20


nbound

m

n

Figure 5. Unstructured grid adjacent to the boundary.

The nal point to be addressed regarding articial dissipation is its treatment on boundaries.This is of importance as it can have a signicant eect on solution accuracy. It is not possibleto evaluate Equation (32) on the boundary as2m requires the use of nodes that lie outsidethe domain. An approximation is, therefore, required and we subdivide treatment into twocases, viz. variables with a normal gradient of change component i.e. d qj=dn=0 and thosewhere the normal gradient is zero.

A typical example of dqj=dn= 0 is velocities on viscous (no-slip) type boundaries. For theapproximation of the dissipation term on the boundary, we again start out by considering aone-dimensional case. The second-order derivative on a boundary may be approximated torst-order accuracy by setting it equal to the value calculated for2 at the adjacent node.This is not expected to introduce a signicant error into the solution as the central dierencescheme described in this paper is at best only rst-order accurate on the boundary in thedirection normal to the boundary. An ecient and straightforward method of extending thisapproximation to multidimensional unstructured grids, is by setting the rst approximate Lapla-cian operator (Equation (36)) at a boundary node (min Figure 5) equal to the value calculatedat the corresponding edge node of the edge closest in the direction to the boundary normal(node n). Although being a simplied version of the approximation advocated by Mavriplis[21], it was found to give good results on all problems attempted to date.

For the case where dqj=dn = 0 e.g. the pressure on a viscous boundary in isothermal ow,a dierent approximation is used. In this work, this very condition is implemented into thecalculation of2 at boundary nodes by only doing computations on edges aligned with the

boundary. Special treatment may be required for highly stretched elements, but this was not

found to be necessary on the problems dealt with in Part II.

4.3. Pseudo-temporal discretization

The pseudo-temporal term in Equation (30) is discretized using an explicit multistagetime-stepping method. The solution is advanced from time t to t+ t with a four-stage



17/20


RungeKutta scheme as follows:

Q0m = Qtm

Qim = Q

0m+it

i11

LGPm (RHSi1

m ) for i =1 to 3

Qt+tm = Q4m

(37)

where the coecients i are taken as 1

4; 1

3; 1

2 and 1. The right-hand side RHSi1m is computed

from Equation (30) as follows:

RHSi1m =

1

m

edges

Fj

i1

mn Gji1

mn

Cjmn+B

jm

m

+ [Si1]m+Dm

(38)

To save computational time, the articial dissipation and viscous terms are updated only atthe rst and third stages.

A local time-stepping approach is employed to accelerate convergence to steady state. In thecase where pseudo-acoustic automation is employed, the local time-step size is calculated usingEquation (18). This equation already takes into consideration both advection and diusioncomponents as described. This is, however, not the case with the standard scheme wherec is prescribed by the user, and a dierent relation is required to account for the diusioncomponent. Consistent with our previous analogy, both advection and diusion componentsare taken into account as follows:

t=

max{max=CFL; |u| +udi=vN} (39)

where max and udi are given by Equations (20) and (22), respectively.In the present work the CFL number is typically set to 2.4 and the von Neumann number

(vN) to 0:5. The exception is natural convection ow where Rayleigh numbers are of theorder 108. Here, vN is reduced by an order of magnitude.

4.4. Real-time temporal discretization

Dual-time-stepping is implemented to eect ecient real-time accuracy. A real time term isadded to Equation (6) as follows:

@W

@t +

@W

@Q

@Q

@t+

@Fj

@xj @G

j

@xj=S (40)

where t and t designate real-time, and pseudo-time, respectively. As before, the above equa-tion is solved by marching in pseudo-time until a pseudo-steady state is reached (@Q=@t

0)

and the real-time-transient system of NavierStokes equations is recovered.The real-time transient term in Equation (40) is discretized in an implicit manner using

a second-order-accurate three-point-backward dierence scheme. The discretized system iswritten as follows:

1 +2

3

tt

m

i1LGPm Qm= it

RHS

i1m

3Wi1 4Wn +Wn+12t

m

(41)



18/20


where Qm=QimQ0m and i =1 to 3. The superscript n + 1 designates the real-time-step

being sought. During the pseudo-time iterations, Wn and Wn1 are held constant with Wi1

being calculated from Qi1 as follows:

Wi1 =

0 0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

Qi1 (42)

When pseudo-steady state is reached and the pseudo-transient terms reach zero the solutionat the next real time-step Wn+1 is calculated from Qt+t . Owing to the implicit real-timenature of the scheme, the allowable real-time step size t is only limited by the requiredtemporal accuracy.

4.5. Boundary conditions

The boundary types used in this work are solid wall, symmetric plane, outow, prescribed tem-perature and adiabatic boundary conditions. The solid wall boundary is imposed byapplying a non-slip condition i.e. the ow velocity is set equal to that of the wall. Planesof symmetry are enforced by setting the normal component of the velocity vector to zero.Outow conditions are applied for internal ows and in this work constitute prescribing a

pressure (typically 0). For non-isothermal problems, adiabatic boundary conditions are im-posed implicitly by setting the normal component of the temperature derivative used in theenergy equation to zero.

5. NOTES ON IMPLEMENTATION

In this section, we list a few notes of interest regarding implementation of the scheme into acomputer code. Although, some points may seem trivial to the experienced, we list them as

pointers for rst time implementers. These are as follows:

C++ was used as the programming language. Although cumbersome in the initial stagesof program development due to time taken to design the code structure, the program-ming system was found to be an object-oriented language that is well suited for writingnumerical code. Care should be taken when using built-in container types as these may

exhibit performance penalties. We circumvented this through specialization [31]. Regarding numerics, dependent variables should be stored in an array of objects contain-ing smaller generic vectors (in our case Q). Operators performing tasks on the vectorssuch as the calculation of derivatives, or articial dissipation, should then be writtenin a manner that is independent of the number of variables contained. This makes forstraightforward extension to analysis types requiring a dierent number of variables.

Edge coecients are stored in one direction only (e.g. in direction m to n).



19/20


Code should be written in such a manner as to minimize square root computations. Theseare several times more CPU intensive than e.g. multiplication and can have a notableeect on code performance.

It is useful to number nodes and edges such that all boundary entities are clustered

together. In this work we assign the rst positions in global arrays to boundary entities.This aids in computationally ecient treatment of boundaries.

6. CONCLUSION

A robust articial compressibility scheme is developed for modelling laminar steady stateand transient incompressible ows over a wide range of Reynolds and Rayleigh numbers.Articial compressibility is implemented in a consistent manner and a method is proposedto calculate generalized preconditioning parameters locally in the interest of convergence rateand robustness. We further attempt to automate AC free parameters for all ow regimes. Anexplicit pseudo-time-marching scheme was employed to facilitate memory eciency. Specialtreatment of the articial dissipation terms is eected in the interest of consistency. A dual-time-stepping scheme was employed which facilitates computational eciency and temporalaccuracy.

In Part II, the developed scheme and variations, viz. preconditioned AC scheme withpredened constant preconditioning parameter, as well as the standard scheme, are appliedto a number of example problems to solve ows over a wide range of Reynolds and Rayleighnumbers. In all cases, the schemes are compared with regard to ease of use, robustness andrate of convergence.

REFERENCES

1. Patankar SV. Numerical Heat Transfer and Fluid Flow. McGraw-Hill: New York, 1980.

2. Chorin AJ. A numerical method for solving incompressible viscous ow problems. Journal of ComputationalPhysics 1967; 2:1226.

3. Belov A, Martinelli L, Jameson A. A new implicit algorithm with multigrid for unsteady incompressible owcalculations. AIAA Paper, 95-0049, 1995.

4. Manzari MT. An explicit nite element algorithm for convection heat transfer problems. International Journalof Numerical Methods for Heat and Fluid Flow 1999; 9(8):860877.

5. Drikakis D, Iliev OP, Vassileva DP. A nonlinear multigrid method for three-dimensional incompressibleNavierStokes equations. Journal of Computational Physics 1998; 146(1):301321.

6. Zhao Y, Zhang BL. A high-order characteristics upwind FV method for incompressible ow and heat transfersimulation on unstructured grids. Computer Methods in Applied Mechanics and Engineering 2000; 190(57):733756.

7. Tamamidis P, Zhang GQ, Assanis DN. Comparison of pressure-based and articial compressibility methods forsolving 3D steady incompressible viscous ows. Journal of Computational Physics 1996; 124:113.

8. Ferziger JH, Peric M. Computational Methods for Fluid Dynamics. Springer-Verlag: New York, 1999.9. Lakshminarayana B. An assessment of computational uid dynamic techniques in the analysis and design of

turbomachinerythe 1990 Freeman scholar lecture. Journal of Fluids Engineering-Transactions of the ASME

1991; 113(3):315352.10. Turkel E. Preconditioning methods for solving the incompressible and low speed compressible equations. Journal

of Computational Physics 1987; 72(2):227298.11. Choi YH, Merkle CL. The application of preconditioning in viscous ows. Journal of Computational Physics

1993; 105(2):207223.12. Weiss JM, Smith WA. Preconditioning applied to variable and constant density ows. AIAA Journal 1995;

33(11):20502057.13. Turkel E. Review of preconditioning methods for uid dynamics. Applied Numerical Mathematics 1993;12:

257284.



20/20


14. Michelassi V, Migliorini F, Martelli F. Preconditioned scalar approximate factorization method for incompressibleuid ows. International Journal of Computational Fluid Dynamics 1996; 7(4):311325.

15. Kwak D, Chang JLC, Shanks SP, Chakravarthy SR. A three-dimensional incompressible NavierStokes owsolver using primitive variables.AIAA Journal1986; 24(3):390396.

16. Lewis RW, Huang HC, Usmani AS, Cross JT. Finite element analysis of heat-transfer and ow problems

using adaptive remeshing including application to solidication problems. International Journal for NumericalMethods in Engineering 1991; 32(4):767781.

17. Nithiarasu P, Zienkiewicz OC. Adaptive mesh generation for uid mechanics problems. International Journal forNumerical Methods in Engineering 2000; 47(13):629662.

18. de Foy B, Dawes W. Unstructured pressure-correction solver based on a consistent discretization of the Poissonequation. International Journal for Numerical Methods in Fluids 2000; 34:463478.

19. Lin FB, Sotiropoulos F. Assessment of articial dissipation models for three-dimensional incompressible owsolutions. Journal of Fluids Engineering 1997; 119:331340.

20. Jameson A, Schmidt W, Turkel E. Numerical solutions of the Euler equations by nite volume methods usingRungeKutta time-stepping schemes. AIAA Paper, 811259, 1981, unpublished.

21. Mavriplis DJ. Accurate multigrid solution of the Euler equations on unstructured and adaptive meshes. AIAAJournal1990; 28(2):213221.

22. Shuen JS, Chen KH, Choi YH. A coupled implicit method for chemical non-equilibrium ows at all speeds.Journal of Computational Physics 1993; 106(2):306318.

23. Liu C, Zheng X, Sung CH. Preconditioned multigrid methods for unsteady incompressible ows. Journal ofComputational Physics 1998; 139(1):3557.

24. Morgan K, Peraire J, Peiro J, Zienkiewics OC. Adaptive remeshing applied to the solution of a shock interactionproblem on a cylindrical leading edge. Computational Methods in Aeronautical Fluid Dynamics. ClarendenPress: Oxford, 1990, 327344.

25. White FM, Viscous Fluid Flow. McGraw-Hill International Editions: Singapore, 1991.26. Rizzi A, Eriksson LE. Computation of inviscid incompressible ow with rotation. Journal of Fluid Mechanics

1985; 153:275312.27. Taove A, Brodwin ME. Electromagnetic scattering problems. IEEE Transactions on Microwave Theory and

Techniques 1975; 23:623.28. Zienkiewicz OZ, Taylor RL. The Finite Element Method Vol. 3: Fluid Dynamics. Butterworth-Heinemann:

Oxford, 2000.29. Cabuk H, Sung CH, Modi V. Explicit RungeKutta method for three-dimensional internal incompressible ows.

AIAA Journal1992; 30(8):20242031.30. L ohner R. Applied CFD Techniques. John Wiley and Sons: New York, 2001.31. Malan AG, Lewis RW, Ransing RS. On C++ object-oriented programming techniques for high-performance

explicit CFD Codes. ECCOMAS Computational Fluid Dynamics Conference Paper 2001.


an improved unsteady, unstructured artificial compressible finite volume scheme for viscous...

Documents