Computer Methods in Applied Mechanics and Engineering 97 (1992) 245-288North-Holland

CMA217

Adaptive implicit/ explicit finite element methodsfor axisymmetric viscous turbulent flows with

moving boundariesW.W. Tworzydlo, C.Y. Huang

The Computarional Mechanics Company, Inc., 7701 North Lamar, Austill, TX 78752, USA

J.T. OdenTexas Institllfe for Computational Mechanics, The Universiry of Texas ar Austin, Austill, TX 78712,

USA

Received 19 August 1991

The extension of the two-dimensional adaptive implicit 1explicit finite elelnent method to axi-symmetric turbulent flow problems is presented. Difficulties involved in the selection of proper flowvariables for the finite element interpolation to avoid singularities and the treatment of complex sourceterms resulting from the axisymmetric formulation are covered in this study. A special data structure isproposed for efficiently implementing the PrandtJ-Van Driest turbulence model with an implicit 1explicit algorithm for adapted unstructured grids. In addition, a moving-grid/eroding-boundaryalgorithm is implemented (specifically, for modeling the burning of a solid propellant) for internal flowanalysis in solid rocket motors. A series of benchmark problems are solved to demonstrate theeffectiveness of the methodology.

1. Introduction

In recent years, a variety of finite element methods have been proposed for high-speedcompressible flow calculations. Many of the popular techniques are based on explicit schemes,the most popular being explicit version of Taylor-Galerkin schemes [1-10] and Runge-Kuttamethods [9, 11], The explicit methods are subject to time step limitations due to stabilityrequirements. This limitation, of course, can be especially restrictive in the analysis ofboundary layers and in low Mach number flows. This reason and others stimulated recentresearch on implicit algorithms, including the SUPG method [12-17], least squares methods[18, 19], implicit version of the Taylor-Galerkin algorithm [20-22] or a Weak StatementTaylor-Galerkin method [23-25].

An attractive approach, introduced in recent years, is the development of implicit/explicitmethods, combining the robust, unconditionally stable implicit schemes with the relativelyinexpensive explicit schemes, to achieve the maximum effectiveness at the minimum computa-tional cost. These methods were initially developed for solid mechanics problem [26,27]. Forcomputational fluid dynamics problems, an implicit/ explicit scheme based on certain versionsof the Taylor-Galerkin' method was developed by Hassan et al. [28]. Similarly, adaptiveimplicit/ explicit approaches for the solution of the incompressible Navier-Stokes equationsusing a vorticity-stream function formulation were presented by Tezduyar and Liou [29,30].Tworzydlo et al. [31] have recently developed a general adaptive implicit/explicit finiteelement method for solving compressible Navier-Stokes equations formulated in Cartesian

0045-7825/92/$05.00 © 1992 Elsevier Science Pt:blishers B.v. All rights reserved

w.W. Tworzydlo et al., Adaptive implicitlexplicir FEM 247

(2.5)

(2.6)

where U, F~ and F~ are vectors of Cartesian components of conservation variables, convectiveflux and viscous flux, respectively, p is the fluid density, p is the thermodynamic pressure, u;are the fluid velocity components, u~ are the velocity components of grid motion, e is thespecific total energy, 7;j are viscous stress components, and qj are components of a heat fluxvector. The viscous stress components, 7jj' for an axisymmetric Newtonian fluid flow can beexpressed as

Au,Til = µ'RU\ 1 + AU2 2 +-----=:. . r

721 = 712 = µ'(U1.2 + U2.1),

(2.7)

where µ. and A are the molecular viscosity and secondary viscosity, respectively, and thelongitudinal viscosity J.LR is defined as µ'R = 2µ. + A. If Stokes' hypothesis is applied, themolecular viscosity and the secondary viscosity are related by A = - ~µ.. The heat fluxcomponents, q;, are given by Fourier's law of heat conduction,

q; = -kT./ ' (2.8)

where k is the thermal conductivity coefficient.For compressible turbulent flows, the mass-averaged Navier-Stokes equations [33] are

solved with appropriate turbulence models to account for the effects due to turbulent eddies.The only complication in the mass-averaged Navier-Stokes equations is to replace themolecular diffusion coefficients, µ. and k, by an effective value accounting for additionalcontributions from the diffusion of turbulent eddies.

3. Implicit/explicit finite element method for compressible flows

•3.1. A general family of implicit Taylor-Galerkin methods

In this section, a general family of implicit Taylor-Galerkin methods is derived. Thederivation presented here is basically an extension of our previous formulation for Cartesiancoordinate systems [~1]. This family is based on a combination of second-order Taylor seriesexpansions in time with a Galerkin approximation in space. Several implicit parameters areintroduced so that, depending on the particular choice, a fully explicit scheme or a variety ofimplicit schemes can be recovered .

Second-order Taylor expansion in timeAssume that the solution u" is given at the time t" and the solution at time t"+ I. is to be

calculated. The value of the solution at ttl and t"+1 can be expressed by the second-orderTaylor expansions around an arbitrary moment t,,+a, where a is the implicitness parameterwith values between zero and one:

U,,+1 = u"+a + (1- a) /:::"t ~n+a + (1- a)2 b.t ii',+a + O(fj"t3) ,

(3.1 )

w. W. Tworzydlo et al., Adaptive implicit/explicit FEM 249

The increment of the first-order time derivative, after linearization, can be expressed as

= (Roo~u .) . + (I'. ilu) - (A ~u) + Q. ~u + t ~u - B ilu .'I .J./ I ./ , ./ I .1 (3.7)

The substitution of spatial derivatives for the second-order time derivative yields theformula

:..:: "v ....c v C .... "'v ....c v cu=[R;j(Fk,k-Fk,k+S -S )).i+[P;(Fu-Fk,k+S -S)L

(AV AC v C] [A(AV AC V C]-[A; Fk,k-Fu+S -S) .i+ Qj Fu-Fu+S -S ).j

+ [t(F~,k - F~.k + SV + SC)] - [B(F~.k - F~.k + SV - SC)] . (3.8)

The consecutive terms in (3.8) involve spatial derivatives up to the fourth order. Limiting thisformula to terms with second-order derivatives, which can be effectively handled by COcontinuous finite elements, yields an approximation,

::.: ... c c ........c ... cU = (A;F k,k),; + (A;S ).; + BF k.k + BS + O( µ., k) ,

~ = (A;AkU,k) ,; + (A;Bu).i + BAku.k + BBu + O( µ., k) ,(3.9)

where O( µ., k) represents symbolically a quantity of the order of viscosity parameters in theNavier-Stokes equations. The increments of inviscid Jacobians A; and B are of the order of~t:

~A; = O(~t), ilB = O(ilt) .

Neglecting these terms in the increment of the second time derivative results in theapproximation

(3.10)

Substitution of (3.3)-(3.7) and (3.9)-(3.10) into the incremental equations (3.2) gives theimplicit non-linear formula: .

~u + a ~t (Ai ~U),i + a ~t B ~u - 'Y ~t (Rij ~U),i

- 'Y 6.t (1'; ~U).i - 'Y ~t Qj ~U,j - Y ~t t ~u1 2 A A AA

- 2(1- 2a)13 At [(A;Ak ~U.k),; + (A;B ilU).; + BAk ~U.k + BB ~u]

= ~t (i;,; - i~,i + SV - SC) + !(1- 2a) ~t2[(Al~.k).; + (AiSC),i + iJi~,k + iJSC]

+ (1- 2a)O(µ., k) ilt2 + (a - y)O(µ., k) M2 + O(M3). (3.11)

This expansion is second-order accurate for the Euler equations, but only first-order accuratefor viscous problems. Notice that a third implicit parameter, 'Y, is introduced in (3.11) tocontrol the implicitness of the viscous terms without affecting the order of accuracy in time[31]. Also note that a fully explicit algorithm is recovered if a = {3 = y = O. A linear stabilityanalysis for the above scheme was presented in [31].

W, W. Tworzydlo et al., Adaptive implicit 1explicit FEM 251

+ !(l - 2a){3 !::.t2[r(AA {j"u)· v· + (AAr . !::.u)· V·I J .J ./ I J .J ,r

+ r(A)? !::.u)· V.; - r(1JAi {j"u)· V - (BA /.j !::.u)· V - r(BB !::.u)· V]} dfl

+ r {a !::.tr(n.A !::.u)· V - 'V !::.t[r(n.R, Ilu)· V + (nR.r .l:1u)· V + r(n,P. !::.u)· V]Jan I I I I IJ • J I IJ . J I I

2 A-!(1- 2a){3 !::.t [r(n;A;Aj !::.u)· V + (n;A;Ajr.j !::.u)· V + r(njA;B l:1u)· v]} ds

= In !::.t[r(F~ - F;)' v.; - (SC - SV). v]

- !(1-2a)!::.t2[r(AA.ll .)·v+(AAr .u)·v., J .J I J .J .1

c ... .... .... c+ AS . V - r(BAu)' V - (BAru)' v - (BS ). v] dfl

I .r } .} J .J

+ f {!::.t rn(F~ - F~)' V + HI - 2a) !::.t2[r(n;A,AJu ,). Vall ' .

+ n;A;A/.ju)· V + (n;A;SC), v]} ds .

A comparison of these two approaches can be found in [35].

(3.14)

Finite element approximationThe Galerkin approximation of the variational problem (3.14) is obtained by partitioning

the domain fl into a finite element mesh and introducing an approximation of trial functionsand their derivatives,

N

u\x, t) = L uI1Jr/x) ,1= 1

N

ll~;(X, t) = L u,1JrI,i(x) ,1= 1

(3.15)

and the same for test function V and radius r. In the above formula, N is the total number ofnodes within the domain fl and 1JrIis the shape function associated with node I. Note thatwhen the implicit/ explicit scheme is combined with adaptive grid strategies, some nodes haveto be constrained in order to enforce the solution continuity across the boundaries of elements[36,37]. For those elements consisting of constrained nodes, the interpolation formula (3,15)needs to be modified according to the method presented in Appendix B. After introducing thisapproximation into the variational statement (3.14), a system of linear equations is obtained,

(M + K)!::.u = R,

where M is the mass matrix, K is the 'stiffness' matrix, and R is the right-hand side vector. It isconvenient to rewrite the above equation in a semi-indexed notation:

(3.16)

where MIJ and KIJ are small blocks of size M x M associated with nodes I and J (M is thenumber of conservation variables, four in this case).

Substituting the derivatives of the radial coordinates, r, t = 0, r.2 = 1, and the definition of(3.6) into (3.14), the particular forms of the contribution to the matrices in the final system(3.16) can be grouped as:(1) Mass matrix:

w.W. Tworiydlo et al., Adaptive implicit 1explicit FEM 253

Artificial dissipationIn order to suppress spurious oscillations of the solution, it is customary to introduce

artificial dissipation. The artificial dissipation is either added explicitly to the governingequations or introduced implicitly by the numerical algorithm. Although the Taylor-Galerkinscheme introduces a certain algorithmic diffusion, it is usually not sufficient to suppressoscillations near shocks. For this purpose, explicit dissipation is usually introduced. Here wefollow the general approach of Tworzydlo et al. [31], in which the artificial dissipation isintroduced as the additional flux to the Navier-Stokes equations (2.1) in the form

;. .....c .....V A A c \'u+F=F+F-S +S ,1,1 1,1 r,l

where the artificial dissipation flux is the function of the solution vector and its derivatives,AA AA A AF =F. (u,u .),

I I .J

with the corresponding Jacobians defined as

A A apARoo = IIJ - au .

·1

A variety of artificial dissipation models [38,39] may be developed by properly defining theartificial fluxes; see [31] for a detailed discussion. Note that to control the implicitness of theartificial dissipation, a fourth implicit parameter, 5, is introduced.

3.2. Implicit / explicit procedures

The family of implicit Taylor-Galerkin methods developed above is the basis for im-plementation of adaptive implicit/ explicit algorithms. These algorithms are discussed in detailin [31]. Here we will review only the most important ideas.

The basic concept of implicit/ explicit algorithms is simple: combine the two methods totake advantage of the superior features of each. The major advantage of the explicit method isthat element computations are relatively inexpensive and simple. The implicit algorithmusually does not suffer from stability limitations of the explicit method and, moreover, allowsfor simple and straightforward control of natural boundary conditions. Its major disadvantageis the much higher cost of element operations and the more complex and expensive solution ofthe resulting system of equations.

An appropriate combination of implicit and explicit algorithms, based on adaptive partition-ing of the domain n, appears to be the most robust and effective approach.

There exists a variety of approaches to actual implementation of implicit/ explicit methods,for example:(1) explicit prediction in the whole domain n, followed by implicit correction for the implicit

zone,(2) partitioning of the domain n into separate implicit and explicit subzones, such that

n(E) n n(1) = 0 and n(E) u n(l) = n,(3) arbitrarily variable implicitness coefficients within the domain n; a = a (x) , etc.A more detailed discussion of these procedures can be found in [31]. In this work, approach(2) was used.

Selection of implicit and explicit zonesThe basic criterion for selection of implicit and explicit zones is simple: for a given time

step. all nodes which violate the stability criterion for an explicit scheme should be treated

Table 1

WW. Tworzydlo et al., AdaptilJe implicitlexplicit FEM 255

Type of Boundary

Supersonic inflowSubsonic inflowSubsonic outflowSupersonic outflowNo-flowSolid wall

IsothermalHeat flux

Euler (not regularized)

4 ess3 ess1esso1ess

Navier-Stokes

4 ess3 ess + 1 nat1 ess + 2 nato ess + 3 nat1 ess + 2 nat

3 ess2 ess + 1nat

rather than the conservation variables. It is important to note that the numbers presented inthe table are true for problems that are not regularized. If artificial diffusion is built into thealgorithm or added explicitly, natural boundary conditions should be imposed on these termseven for Euler problems. Moreover, since artificial diffusion (in contrast to the naturalviscosity) affects all the conservation variables, the number of natural boundary conditions forthese terms should actually be one more than for the (non-regularized) Navier-Stokesequations.

The details of implementation of basic boundary conditions (1)-(3) are discussed in thecontext of implicit / explicit algorithms in [31]. We present next the formulation and implementation of the porous/burning wall boundary conditions.

4.1. Porous wall with a prescribed mass injection rate

A typical application of this type of boundary condition is to model the burning surface of asolid propellant with simultaneous boundary motion. A total number of four quantities areprescribed (see Fig. 2). These quantities are the tangential and normal velocities of the wall(uTW and uNW), the mass flux across the wall (mN), and the temperature of the wall (T w). Thecorresponding boundary conditions are:(a) The tangential velocity of the fluid is equal to the tangential velocity of the wall,

UT = UTW·

(b) The balance of the net mass flux is satisfied:

p(UN - uNW) = mN (normal momentum) .

(c) The temperature of the fluid is equal to the temperature of the wall,

Fig. 2. Porous wall boundary conditions.

w. W. Tworzydlo et al., Adaptive implicit 1explicit FEM 257

Following the same procedure as before, with test function defined as 8mN, we obtain thecorresponding kernel, k, and the right-hand side, r, as

where

1 A

k=-d0d,t::

1 [ - (n " - )]dAr = - mN - mN - P UNW ,t::

(4.10)

(4.11)

(4.12)

and nj, i = 1,2 denote the components of the outward unit normal vector.For condition (c), we apply the same penalty procedure to enforce the prescribed

temperature. The resulting penalty term is (tested with the variation of total energy)

d=[O,O,O,l].

The temperature can be expressed in terms of conservation variables as

(EI mimi)T= y(y-1) P - 2p2 .

Then the familiar vectors d = aT/ au and d = aE) av can be derived as

_ [ EI E k - U 1 - Uz 1]d=y -2+2'-'-'- ,

p p p p p

The corresponding kernel of the stiffness matrix and the right-hand side are

(4.13)

(4.14)

1 (- ,,)Ar=-Tw-T d.t::

5. Turbulence modeling

In the present work, an algebraic turbulence model based on the Prandtl mixing lengththeory was selected and implemented in the context of adaptive implicit / explicit methods.This model is simple in formulation and the eddy viscosity can be calculated without solvingadditional equations. However, it is rather difficult to implement for unstructured meshes usedin adapted grids and for complicated geometries. To overcome these difficulties, a datastructure is proposed so that the eddy viscosity can be efficiently calculated. This sectionpresents details of the numerical implementation of the Prandtl-Van Driest turbulence modelwhich is suitable for internal flow analysis.

5.1. Prandtl-Van Driest turbulence model

In the Prandtl-Van Driest turbulence model, the turbulent eddy' viscosity is calculatedaccording' to

I-Lj = plwl/z ,

where p is a local fluid density, Iwl is a local magnitude of vorticity, and I is a Prandtl mixinglength. The Prandtl mixing length can be obtained from the formula

WW Tworzydlo et al., Adaprive implicitlexplicit rEM

o~t~ 1.0

Fig. 3. A length scale of an interior node is a valid projection with respect to a viscous boundary.

259

Each node has at least one length scale associated with a viscous boundary (Fig. 5).With the above functional specifications and criteria in mind, we present a data structure for

efficiently implementing a Prandtl-Van Driest turbulence model.The storage required in this data structure is defined by the following arrays:

(a) Real arrays:RHOWALL(2*MXBCD),TAUWALL(2*MXBCD),CMUWALL(2*MXBCD),RTZDAT(2,2*MAXND),TLENX(MAXND)

(b) Integer arrays:NVISPAN,NVISNOD,NODVPAN(4,2*MXBCD),NLENOD(MAXND),ITZDAT(2*MAXND),ITZPTR(MAXND)

INTERIOR NODE

VISCOUS BOUNDARY

Fig. 4. A valid projection breaks the boundary,

w. W. Tworzydlo et al., Adaptive implicit 1explicit FEM 261

and can be identified by four integers: the element number it belongs to, the side number, andthe node numbers (with respect to total number of viscous nodes) of its two end points. ArrayNODVPANis used to store this information. The reason for storing the node numbers of theend points for each viscous panel is that they could be readily used for calculating y + for eachinterior node by using interpolation.

The second set of data is utilized to store information that is used to calculate the Prandtlmixing length for each node during the solution procedure. This set of data consists of apointer, the number of length scales, and the values. of the length scales associated with anode. For each length scale, the associated viscous panel number and the local coordinate ofthe projection are also stored. A pointer array designed to take care of variable length scales,to efficiently allocate the storage and to allow direct access to the data for the calculation ofthe length scale is used.

Summary of storage(1) A total of 7 * MAXNDwords are required for storing information associated with the length

scales for each node (assume averaged two length scales for each node).(2) A total of 3 * M){VISP words are required for storing solutions on the solid (no-slip) walls

for efficiently interpolating solutions.For a grid system with MAXND= 2500, 140 kbytes of storage are required (assuming an

average of two length scales for each node). This storage requirement is comparable to thestorage required by the data structure designed by Rostand [33]: 90 kbytes for the same gridsize and a 'single' length scale for each node. However, it must be noted that Rostand's datastructure was designed for implementing both an inner layer and an outer layer turbulencemodel and for simple geometries only (convex geometry was assumed).

6. Moving grid algorithm

The porous wall boundary condition discussed in Section 4 can be applied to model thephysical process of the burning of a solid propellant. Another computational issue which mustbe addressed is associated with the fact that as the solid propellant burns, the surface of thesolid propellant is simultaneously eroded. Computationally, this process can be modeled byeroding boundary algorithms, but arbitrary boundary movement may cause unacceptablemesh distortions for elements near the moving boundary. Therefore, the influence of theboundary movement on the interior nodes must be consistently monitored to avoid excessivelylarge element distortions. The moving grid algorithm we propose here to achieve theseobjectives can be stated as follows.

For a given computational domain, fl, anI represents a part of the boundary which issubjected to a prescribed boundary motion and an2 represents the stationary component, suchthat an = anI U an2 (see Fig. 6).

To avoid large mesh distortions due to the boundary motion, the interior elements areallowed to stretch. The grid velocities Uk for the stretching are calculated by solving aboundary-value problem defined by the Laplacian operator (which has certain smoothingproperties):

"V2Uk = 0 in n, k = 1,2 ,

where Uk represents the Cartesian components of grid velocity and UN is a prescribed normal

w.W. Tworzydlo et al., Adaptive implicir 1explicit FEM 263

7.1. Supersonic flow over a wedge / cone

The purpose of solving this test case is twofold. First, it is used to verify the axisymmetricformulation by comparing results with an exact solution. Second, it is used as a benchmarktiming test for planar and axisymmetric problems with the same flow conditions. A uniformsupersonic flow of M = 2.0 over a cone with the semi-vertex angle 15° is considered. For thetiming test, the planar counter example (flow over a wedge) is also solved.

The analytic solution for the shock angle is 45.5° for the planar wedge flow and 33.8° for theaxisymmetric conical flow [51]. Good agreement with the analytic solution can be observedfrom the plots shown in Fig. 7. The fan-like Mach contours plot in the axisymmetric case (Fig.7(b)) is in agreement with the conical flow theory [52], according to which the flow conditions

(a)

(b)

(c)

Fig. 7. A supersonic flow (M~ = 2) over a cone and a wedge. (a) Adapted grid, (b) Mach contours. (c) streamlines.

w.W. TlVorzydlo et al .. Adaptive implicit 1explicit FEM

Gi1::J implicit

_ eHplicit

Fig. 9. Implicit/explicit zones used for solving conical nozzle problem.

265

olid I 111:"1~~

2.8 3,675

Fig. 10. Mach contours in a conical nozzle show reflected shock pattern.

waves. This pattern is best illustrated by the Mach number contours shown in Fig. 10. Thehighly curved sonic line shown in Fig. 11 demonstrates the two-dimensional effect of this typeof nozzle flow. The comparisons of experimental data [54], numerical results from Serra [53],and our prediction for the Mach number distribution along the nozzle centerline and nozzlewall are presented in Fig. 12. Excellent agreement with the experimental data can beobserved.

7.3. lnviscid supersonic flow through a two-dimensional channel

An inviscid supersonic flow of Moe = 3.0 through a two-dimensional channel is analyzedusing both structured and unstructured grids. The geometry of this problem is taken from [55].The grids were adapted to the third level. The final adapted grids consist of 5254 nodes and6228 elements for a structured mesh, and 7946 nodes and 8835 elements for an unstructuredmesh. The adapted grids and pressure contours for both cases are shown in Figs. 13 and 14.Since both the structured and unstructured meshes were dominated by small elements ofrelatively uniform size, the most economical approach (chosen automatically by the costminimization option) was to solve these problems with a fully explicit algorithm. A verycomplex wave interaction pattern in the channel is predicted by the code. All reflected shockwaves shown in the pressure contours arc weak shocks. With the unstructured mesh, arelatively better shock resolution was obtained at the expense of 50% more degrees of

w.W. Tworzydlo et al., Adaptive implicit 1explicit FEM 267

Fig. 13. A supersonic flow (M~ = 3) through a channel. structured grid. (a) Adapted grid, (h) pressure contours.

Fig. 14. A supersonic flow (M~ = 3) through a channel, unstructured grid. (a) Adapted grid, (b) pressure contours.

The internal wave pattern can be seen clearly from the adapted grid and the streamlinesshown in Fig. 15. As in the previous case, the least expensive time marching algorithm for thisgrid is the fully explicit one. By comparing the Mach contours plot, Fig. 16(a), with theprevious test case, it is interesting to note that the regular reflection of the shock wavesobserved in the previous test case becomes a Mach reflection which is caused by a largeincidence of impinging shock angle.s to a solid wall [52]. One can observe that the three-dimensional relieving effect is no longer valid for the internal flow.

In addition to the Mach reflection, several triple points can be seen in the Mach contoursplot. A triple point is defined as the intersection of an impinging shock wave, a Mach stem(normal shock), and the reflected shock wave. It is well known in gas dynamics theory thatflow across shock waves of different strengths create different entropy changes. Therefore, at

WW Tworzydlo er al., Adaptive implicitlexpJicit FEM 269

II;;:;'J~5

o. . 1lIr:'.. tI.~,

3.75

b)

1 Il:~t'a~16 7.875

3.25

C)

Fig. 17. A supersonic flow (Mx = 3) through an axisymmetric tube. (a) Density contours, (b) pressure contours, (c)Mach contours.

bow shock is approximately s / R = 1.56 for both structured and unstructured meshes. Thesepredicted bow shock profiles agree very well with the experimental observations (57].

7.6. Weakly compressible viscous flow over a sphere

The flow conditions considered are M~= 0.1 and Re = 100.0. The surface of the sphere wasmodeled as a no-slip isothermal wall. The axis of symmetry was modeled by a nq-flow orno-penetration boundary. Subsonic inflow/ outflow boundary conditions were imposed on therest of the artificial boundaries with the number of prescribed boundary conditions automati-cally adapted to the flow pattern [31]. No artificial dissipation was added for this case. Thecomputational domain was first discretized by using a mixed structured/unstructured grid.During the solution the grid was adapted to the first level in the regions close to the solidboundary and in the separation region. The final grid consists of 2535 nodes and 2457clements.

W, W. Tworzydlo ct al., Adapril'e i;"plicit / explicit FEM

(a)

271

Fig. 19, A weakly compressible viscous flow (M~ = 0.1. Re = 100) over a sphere. (a) Adapted grid, (b) closeupview of adapted grid.

' .• implici.t

/_ eHplicit

Fig. 20. A weakly compressible flow (M~ = Il.l. Re = 1(0) over a sphere, implicit/explicit zones.

w.W. Tworzydlo et al., Adaptive implicit 1explicit FEM 273

(a) LEGENDo RDRP12Do W00

o

u'O

We0::::>U'lU'lWo0::0..0

u'O

o,

o

-lHE1R X IOu-2

°~T (b)

°CD

0

'">-I--WO-1-"-a::lSl> 0

N

0

e

C)

'",

LEGENDo RDRP12Do R I M0N/CHEN6 W~B

,80 2.00

THETR X IOu-2

Fig. 22, A weakly compressible flow (M~ = 0.1, Re = 100) over a sphere. (a) Comparison of surface pressuredistribution, (b) comparison of surface vorticity distribution.

small secondary separation bubble was predicted by the code. Similar flow phenomena wereobserved in an experimental work performed by Sandborn [63].

The comparisons of our results with the flow predicted by INS3D code [60] for thelongitudinal velocity distributions at several axial locations are shown in Figs. 25 and 26. Goodagreement can be observed for those locations far away from the separation bubble. However,relatively large discrepancies can be seen in the regions close to the separation bubble. Thesedifferences can be caused by different turbulence modeling for this flow because, as pointedout by Monson et al. [60], turbulence closure models have been developed and optimized forexternal flows. Their importance and choice for internal flows with strong curvature stillremains to be established. Further numerical studies are required to investigate this dif-ference.

~~~<5...~~~:;:,:-

~

:l>.f}~~.<II

~.-§:---<II

~§:

(dl

c(el

Fig. 24. Internal flow in a turnaround duct, Re = 106, M~ = 0,1. (a) Velocity vectors. (b) stream function contours,

(c) pressure contours, (d) vorticity contours.N-.II.JI

w. W. TlVorzydlo et al.. Adaptive implicit 1explicit FEM 277

LECENO0 RE=\O .. S0 RE= 10 .. 6

(a) 6 RORPT.RE=ES0 RORPT .RE=ES

I ~N-0

-(IJ

alL.WID<r:::::lO"-J ...

aN

a°a, I O. 10 0.20 0.30 O. ~O 0.50 0.60 0.70 0.60 0.90 1. 00

0I

Y/H

LECENO0 RE=10 •• Sa RE=lO .. S

N (b) A AOAPl. RE=E5~I 0 AOAP1 • RE=ES

~ so:-= -===--------0

-(IJ

au..WID0::::::l0"-J ...

ci

N

a

°o.0.50 0.60 0.70 0.80 0.90 t. 00O. JO 0.20 0.30 O.~O

Y/H

Fig. 26. Internal flow in a turnaround duct, comparison of longitudinal velocity distribution. (a) xl H == 4.0, (b)xlH == 12.0.

..Fig, 27. Internal flow in a porous cylinder with a nozzle, Re == 8.0 x 105

, mass injection rate == 0.0018. (a) Initialgrid, (b) stream function contours.

w. W. Tworzydlo et al., Adaptive implicill explicit FEM 279

~1 ~[(b)

~:\~\~Y":::::::=X~:'-~

~-~-"'="'d~~~---==·~---~~===-=-o- -_,,-~t~-·=

Fig. 29. Internal flow in a slotted chamber with a nozzle, Re = 8.0 x 105, mass injection rate = 0.0027. (a) Initial

grid, (b) pressure contours.

[65] for detailed geometry specifications. The initial grid consists of 70 x 20 elements in thecylindrical part, 30 x 20 elements in the nozzle section, and 10 x 12 elements in the inhibitorregion. The surfaces of the inhibitor are treated as a porous wall. The normalized massinjection rate is 0.0027 and the Reynolds number is 8.0 x 105

. The mass injection rate is heldconstant for all porous walls. The initial condition and the rest of the boundary conditions areexactly the same as for the previous test case.

Figures 29(a, b) show the initial grid and the pressure contours plot, respectively. Thepressure contours plot indicates that the chamber pressure is nearly constant and a rapid flowexpansion occurred in the divergent section due to the large nozzle contraction. A very thinvelocity boundary layer along the nozzle wall can be seen from the closeup view of the velocityvectors plot and Mach contours plot in Fig. 31. The adaptively selected implicit/ explicit zonesin the nozzle section are shown in Fig. 30. As expected, implicit elements (670 out of 2453)are clustered near the solid wall where the element size is the smallest. The computations withthis algorithm were about four times less expensive than with the fully explicit algorithm or thefully implicit algorithm. The comparisons of the normalized axial velocity distribution atseveral axial 10catioI"s with the data given by [65] are shown in Fig. 32. Generally speaking,the velocity boundary layer predicted by our code is relatively thinner. This difference may be

r_ implicit

_ eHPI}Ci~

Fig. 30. Internal flow in a slotted chamber with a nozzle. Re = 8.0 x 105, mass injection rate = 0.0027. Implicit 1

explicit zones in the nozzle section.

CD

0

eo0

:x0::"-0:: ~

a

N

0

~ lIb)0

o

CD

a

eoc::i

:x0::"-0:: ~

c::i

N

c::i

~ lIe)0

o

CD

c::i

eoc::i

:x0::"-a:: ...

a

N

0

~ lId)0

0.20 O,QO 0.60 0.80N0RHRLIZEO RXIRL VEL0CITY

1. 00 1.20

lECfHOo ElP. OAT~• MIMT.. ADAf'TlD

::E::E~c~~""-c~~:l:.""-¥.iii-.3~§:--..~

-g.-2:"'l~~

0.20 0.40 0.60 0.80 1.00 1.20 0.10 0,20 0.30 O.QO 0.50 0.60 0.70 0.80 0.90 1.00N0RMRLIZEO RXIRL V£L0CITY N0RHRLIZED RXIRL VEL0CITY

Fig. 32. Internal flow in a slotted chamber with a nozzle. Re = 8.0 x lOs, mass injection rate = 0.0027. Comparisonof axial velocity distribution. (a) zID=2.9L (b) zID=4.09. (c) zID=5,33. (d) zID=6.51.

N00-

w.W. Tworzydlo el at., Adaptive implicit 1explicit FEM

where

283

)1=,,-1,

and " is the ratio of specific heats.(2) The Jacobian matrix due to viscous source terms, t, is defined as

where

00

A 2Aml2ml.1-~PI

T1=1 P p'A 2Am2

+ 2 m-;..2 - ~ P.2P P

0

0 0 0 00 0 0 0

T = 12µm2 0-2µ

02 _2

P P0 0 0 0

oo

o 0o 0

Ap2 P.2 0

o 0

I apvp =--.!.

ail

(3) The Jacobian matrices due to viscous source terms, Q, are defined as

• asv 1 . asv 1QI = au = -;.QI' Q2 = au = -;.Q2

.1 .2

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0

QI =1 AmI -A0 0 Q2= Am2 0

-A0- , -2 p2P P P

0 0 0 0 0 0 0 0

(4) The viscous Jacobians, pi = aP~ / au, are defined as

o 000PI pI I

21 22 P23 0

PI pt I31 32 P 33 0

P~1 P~2 P~3 P~4where the matrix components of p! are given by

I 1 ( mt P,I m2p'2)P21=2 -µRmll-Am22+2µR-+2A- ,P . , P P

pI µR I A22= - '2 PI' P 23 = - 2 P 2 'P , P .

I µ ( P.2 P.I ml)P31 = 2 -m2 1- ml2 +2ml - +2m2 - + - ,P . . P P r

w.W. Tworzydlo et al.. Adaptive implicit/explicit FEAt 285

A typical situation is shown in Fig. 34. The notion of constrained approximation is introducedby the requirement that the global approximation across the boundaries of elements must becontinuous [36,37]. This implies that these hanging nodes will not be considered as thedegrees of freedom for a problem; rather, they are constrained by some given degrees offreedom and require special treatment in the global approximation (eq. (3.15»). By knowingthe constrained relations, however, the modification of the global approximation for theconstrained nodes can be implemented in an element-by-element fashion.

Suppose that one is interested in performing element calculations for the element NELshown shaded in Fig. 34. The solution within this element can be expressed as

4

,/ = ~l UNODES(/.NEL) 'IJf/ = U I 'IJf, + u2 'lJf2 + U3 'lJf3 + U4 'lJf4 • (B.l)

where 1JJj is the local shape function associated with node I, and u/ is the solution vector atnode I. However, nodes 1 and 3 are hanging nodes and the numerical solution at these twonodes for bilinear elements are constrained by

(B.2)

This constrained relation is valid based on the assumption that bisecting an element is used asthe rule for grid refinement. Substituting (B.2) into (B.l) and rearranging terms yields

'IJf\ 1Jf., ( 'IJf\ 'IJf, ) (B 3)ue

= U9 2 + U2 'lJf2 + Us 2 + U4 2 + 2 + 'lJf4 . .

By defining new shape functions, <p/, as

(B.4)<P3 = !'lJf3 ' <P 4 = 0 'lJf1 + ~'lJf3 + 'lJf4) ,

and defining (u9, u

2' us' u

4) as the physical degrees of freedom associated with this element,

then the solution vector within element NEL can be interpreted as

where

4

ue= 2: UNODES(/.NEL)<P/ ,

/=\

NODES(I,NEL) = 9,NODES(3,NEL) = 8 ,

NODES(2,NEL) = 2 ,NODES( 4,NEL) = 4 .

(B.5)

6

Fig. 34, Elements consisting of constrained nodes.

w. W. Tworzydlo et al., Adaptive implicit 1explicit FEM

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[15] TJ .R. Hughes. M. Mallet and A. Mizukami, A new finite element formulation for computational fluiddynamics: II. Beyond SUPG, Comput. Methods App!. Mech. Engrg. 54 (1986) 341-355.

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aerodynamics, AIAA Paper 89-0656, AIAA 27th Aerospace Sciences Meeting. Reno, NV, 1989.[25] G.S. Ianelli and A.J. Baker, An efficient solution-adaptive implicit finite element CFD Navier-Stokes

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viscous high speed flows, AIAA 27th Aerospace Sciences Meeting, Reno, NV, 9-12 January 1989.[29] T.E. Tezduyar and J. Liu, Element-by-element and implicit/explicit finite element formulations for computa-

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