A multiscale finite element method for the incompressible Navier–Stokes equations A. Masud * , R.A. Khurram Department of Civil and Materials Engineering, The University of Illinois at Chicago (M/C 246), 842 W. Taylor St., Chicago, IL 60607-7023, USA Received 24 October 2004; received in revised form 5 March 2005; accepted 26 May 2005 This paper is dedicated to Professor Thomas J.R. Hughes on the occasion of his 60th birthday Abstract This paper presents a new multiscale finite element method for the incompressible Navier–Stokes equations. The proposed method arises from a decomposition of the velocity field into coarse/resolved scales and fine/unresolved scales. Modeling of the unresolved scales corrects the lack of stability of the standard Galerkin formulation and yields a method that possesses superior properties like that of the streamline upwind/Petrov–Galerkin (SUPG) method and the Galerkin/least-squares (GLS) method. The multiscale method allows arbitrary combinations of interpolation func- tions for the velocity and the pressure fields, specifically the equal order interpolations that are easy to implement but violate the celebrated Babuska–Brezzi condition. A significant feature of the present method is that the structure of the stabilization tensor s appears naturally via the solution of the fine-scale problem. A family of 2-D elements comprising 3 and 6 node triangles and 4 and 9 node quadrilaterals has been developed. Convergence studies for the method on uni- form, skewed as well as composite meshes are presented. Numerical simulations of the nonlinear steady and transient flow problems are shown that exhibit the good stability and accuracy properties of the method. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Navier–Stokes equations; Multiscale finite elements; Stabilized methods; HVM formulation; Arbitrary pressure–velocity interpolations 0045-7825/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2005.05.048 * Corresponding author. Tel.: +1 312 996 4887; fax: +1 312 996 2426. E-mail address: [email protected](A. Masud). Comput. Methods Appl. Mech. Engrg. 195 (2006) 1750–1777 www.elsevier.com/locate/cma
A multiscale finite element method for the incompressibleNavier–Stokes equations
A. Masud *, R.A. Khurram
Department of Civil and Materials Engineering, The University of Illinois at Chicago (M/C 246),
842 W. Taylor St., Chicago, IL 60607-7023, USA
Received 24 October 2004; received in revised form 5 March 2005; accepted 26 May 2005
This paper is dedicated to Professor Thomas J.R. Hughes on the occasion of his 60th birthday
Abstract
This paper presents a new multiscale finite element method for the incompressible Navier–Stokes equations. Theproposed method arises from a decomposition of the velocity field into coarse/resolved scales and fine/unresolvedscales. Modeling of the unresolved scales corrects the lack of stability of the standard Galerkin formulation and yieldsa method that possesses superior properties like that of the streamline upwind/Petrov–Galerkin (SUPG) method andthe Galerkin/least-squares (GLS) method. The multiscale method allows arbitrary combinations of interpolation func-tions for the velocity and the pressure fields, specifically the equal order interpolations that are easy to implement butviolate the celebrated Babuska–Brezzi condition. A significant feature of the present method is that the structure of thestabilization tensor s appears naturally via the solution of the fine-scale problem. A family of 2-D elements comprising 3and 6 node triangles and 4 and 9 node quadrilaterals has been developed. Convergence studies for the method on uni-form, skewed as well as composite meshes are presented. Numerical simulations of the nonlinear steady and transientflow problems are shown that exhibit the good stability and accuracy properties of the method.� 2005 Elsevier B.V. All rights reserved.
The standard Galerkin formulations for the incompressible Navier–Stokes equations suffer from severalnumerical deficiencies that are now well documented in the literature. First, the convection term needs spe-cial attention, and second, the practically convenient combinations of interpolation functions for the velo-city and pressure fields often do not work. In order to correct this deficiency in the standard Galerkinapproach, Hughes and colleagues introduced the streamline upwind/Petrov–Galerkin (SUPG) method[9]. The SUPG method turned out to be the forerunner of a new class of stabilization schemes, namelythe Galerkin/least-squares (GLS) stabilization methods [23,24]. In the GLS method least-squares formsof the residuals that are based on the corresponding Euler–Lagrange equations are added to the Galerkinfinite element formulation. These residual based terms are defined over the element interiors only, and theterms on the element boundaries are excluded. This preserves the continuity requirements in the C0 ap-proach to the problem. In the context of the advection-dominated diffusion phenomenon, it leads to (i) sta-bilization of the advection operator without upsetting consistency or compromising accuracy, and (ii)circumvention of the BB (inf–sup) condition that restricts the use of many convenient interpolations.The underlying philosophy of the SUPG and the GLS methods is to strengthen the classical variationalformulations so that the discrete approximations, which would otherwise be unstable, become stable andconvergent. The GLS stabilization was soon followed by the unusual stabilized methods introduced byFranca and coworkers [11,12]. Concurrently, another class of stabilized methods that is based on the ideaof augmenting the Galerkin method with virtual bubble functions was introduced by Brezzi and coworkers[4–8]. For a review of the application of stabilized methods in computational fluid dynamics, see[3,10,13,16–19,25–27,30,32,34–38] and references therein.
In the mid-90s Hughes revisited the origins of the stabilization schemes from a variational multiscaleview point and presented the variational multiscale method [20,22]. In this method the different stabilizationtechniques come together as special cases of the underlying subgrid scale modeling concept. Employing thismethod, hereon termed as the Hughes variational multiscale (HVM) method, Masud and co-workers devel-oped multiscale/stabilized formulations for the linearized incompressible Navier–Stokes equations [28], theadvection–diffusion equation [32], the convective–diffusive heat transfer [1], the Darcy flow equations [31],and the Fokker–Planck equation [29].
Extending this idea to the nonlinear regime, this paper presents a new multiscale finite element methodfor the incompressible Navier–Stokes equations. We know that the discretization of the computational do-main introduces an artificial length scale in the solution procedure, and the features in the solution that arefiner than this length scale are poorly approximated. The ensuing lack of accuracy of the computed solutionis one of the contributing factors that lead to the poor stability properties of the standard Galerkin methodin several important cases. In the multiscale finite element method presented here, the decomposition of thevelocity field into coarse and fine-scales and an introduction of a model for the fine-scale velocity, correctsthe lack of stability of the standard Galerkin variational formulation of the Navier–Stokes equations. In theproposed multiscale method, the fine-scale part of the velocity field is modeled via residual based terms thatare derived consistently. A substitution of the fine-scale solution in the coarse-scale problem results in amathematical nesting of scales. Consequently, the proposed formulation inherently possesses higher accu-racy on cruder discretizations. The consistency of the formulation coupled with the enhanced accuracy andstability, yields a method with very good convergence properties. Another significant feature of the presentmethod is that the structure of the stabilization tensor s appears naturally via the solution of the fine-scaleproblem.
An outline of this paper is as follows. Section 2 presents the strong and weak forms of the incompressibleNavier–Stokes equations. Section 3 presents the HVM method, and a detailed exposition on the develop-ment of the new multiscale finite element method for the incompressible Navier–Stokes equations. Section 4presents numerical results, and conclusions are drawn in Section 5.
Let X � Rnsd be an open bounded region with piecewise smooth boundary C. The number of spacedimensions, nsd, is equal to 2 or 3. The incompressible Navier–Stokes equations are written as follows:
_vþ v � rv� 2mr � eðvÞ þ rp ¼ f in X��0; T ½; ð1Þdivv ¼ 0 in X��0; T ½; ð2Þv ¼ g on Cg��0; T ½; ð3Þr � n ¼ ð2mrsv� p1Þ � n ¼ h on Ch��0; T ½; ð4Þvðx; 0Þ ¼ v0ðxÞ on X0; ð5Þ
where v is the velocity vector, p is kinematic pressure, f is the body force vector, m is kinematic viscosity, $svis the symmetric part of the velocity gradient, and 1 is the identity tensor. e(v) is the strain rate tensor whichis defined as
eðvÞ ¼ 12ðrvþrvTÞ.
Eqs. (1)–(5) represent balance of momentum, the continuity equation, the Dirichlet and Neumann bound-ary conditions, and the initial condition, respectively.
2.1. The Galerkin form
Let w and q represent the weighting functions for velocity v and pressure p, respectively. The appropriatespaces of weighting functions for velocity and pressure are
V ¼ fwjw 2 ðH 1ðXÞÞnsd ; w ¼ 0 on Cgg; ð6ÞP ¼ fqjq 2 L2ðXÞg. ð7Þ
We now define the corresponding time dependent spaces of functions Vt and Pt for the time dependentvelocity and pressure trial solution, respectively.
Vt ¼ fvð�; tÞjvð�; tÞ 2 ðH 1ðXÞÞnsd ; vð�; tÞ ¼ g on Cg��0; T ½g; ð8ÞPt ¼ fpð�; tÞjpð�; tÞ 2 L2ðXÞg. ð9Þ
The classical weak formulation of (1) and (2) is: Find v 2 Vt, p 2 Pt, such that, for all w 2 V, q 2 P,
where ð�; �Þ ¼RXð�ÞdX, i.e., the L2 product of the indicated arguments over domain X.
Remark 1. This formulation has served as the basis of the Galerkin finite element method. It is known thatonly certain combinations of the velocity and pressure interpolations are stable [23]. In the sequel we developa weak form which is inherently more stable, and accommodates a larger variety of stable interpolations,such as equal-order continuous interpolations that are known to be unstable in the classical formulations.
3. Hughes variational multiscale method
3.1. Multiscale decomposition
We consider the bounded domain X discretized into nel nonoverlapping regions Xe (element domains)with boundaries Ce, e = 1,2, . . .,nel, such that
We denote the union of element interiors and element boundaries by X 0 and C 0, respectively.
X0 ¼[nele¼1
ðintÞXe ðelement interiorsÞ; ð13Þ
C0 ¼[nele¼1
Ce ðelement boundariesÞ. ð14Þ
We assume an overlapping sum decomposition of the velocity field into coarse-scales or resolvable scalesand fine-scales or the subgrid scales. Fine-scales can be viewed as components associated with the regions ofhigh velocity gradients.
vðx; tÞ ¼ �vðx; tÞ|fflffl{zfflffl}coarse scale
þ v0ðx; tÞ|fflfflffl{zfflfflffl}fine scale
. ð15Þ
We assume that v 0 is represented by piecewise polynomials of sufficiently high order, continuous in x butdiscontinuous in time. In particular v 0 is assumed to be composed of piecewise constant-in-time functions.Accordingly, we have
vðx; tÞ ¼ �vðx; tÞ þ v0tðxÞ.
Thus,
_vðx; tÞ ¼ _�vðx; tÞ. ð16Þ
Likewise, we assume an overlapping sum decomposition of the weighting function into coarse and fine-
scale components indicated as �w and w 0, respectively.
wðxÞ ¼ �wðxÞ|ffl{zffl}coarse scale
þ w0ðxÞ|fflffl{zfflffl}fine scale
. ð17Þ
We further make an assumption that the subgrid scales although nonzero within the elements, vanish iden-tically over the element boundaries.
v0tðxÞ ¼ 0 on C0t; ð18Þ
w0 ¼ 0 on C0. ð19Þ
We now introduce the appropriate spaces of functions for the coarse and fine-scale fields and specify adirect sum decomposition on these spaces.
Vt ¼ Vt �V0t; ð20Þ
where V in (20) is the space of trial solutions and weighting functions for the coarse-scale velocity field andis identified with the standard finite element space defined in (6),
�vðx; tÞ 2 Vt � C0ðXÞ \Vt; �vðXeÞ ¼ PkðXeÞ; ð21Þ
where PkðXeÞ denotes complete polynomials of order k over Xe.On the other hand, various characterizations ofV0
t are possible, subject to the restriction imposed by thestability of the formulation that requires Vt and V0
t to be linearly independent. Consequently, in the dis-crete case V0
t can contain various finite dimensional approximations, e.g., bubble functions or p-refine-ments, that satisfy (18).
Remark 2. The pressure field can also be decomposed into coarse and fine-scales. However, without loss ofgenerality we assume that the fine-scale pressure field is zero. This assumption helps in eliminating theunnecessary terms that would otherwise emanate from the fine-scale part of the weak form of the continuityequation (11).
3.2. The multiscale variational problem
We now substitute the trial solutions (15) and the weighting functions (17) in the standard variationalform (10) and (11), and this becomes the point of departure from the conventional Galerkin formulations.
With suitable assumptions on the fine-scale field, as stipulated in (19), and employing the linearity of theweighting function, we can split the problem into coarse and fine-scale parts.
The coarse-scale subproblem W and the fine-scale problem W0 can be written as follows:The coarse-scale problem
The coarse and fine-scale equations are in fact nonlinear equations wherein the nonlinearity is engen-
dered by the convection term. In general, to solve nonlinear equations we need to linearize them. To keepthe linearization process simple, we employ ideas from the fixed point iteration method that yields thefollowing linearized formulations for the coarse and fine-scale problems indicated as W and W0,respectively.
In (29) and (31), vc is the last converged solution from the fixed point iteration. It is important to note thatin W0 the weighting function slot only contains the fine-scale weighting functions. The general idea at thispoint is to solve the fine-scale problem, defined on the sum of element interiors, to obtain the fine-scale solu-tion v 0. This solution is then substituted in the coarse-scale problem given by (29) and (30), thereby elim-inating the explicit appearance of the fine-scales v 0 while still modeling their effects.
Let us consider the fine-scale part of the weak form W0, which, because of the assumption on the fine-scale space, is defined over X 0. Employing linearity of the solution slot in Eq. (31), and rearranging terms,we get
where �r ¼ f � _�v�rp � vc � r�vþ 2mD�v. We have applied integration by parts, and condition (19) on w 0, tothe fifth integral on the right-hand side of (32) to arrive at (33).
Our objective at this point is to solve (34) either analytically or numerically to extract the fine-scale solu-tion v 0 that can then be substituted in the coarse-scale problem W. This would eliminate the explicit depen-dence of (34) on v 0, while the ensuing terms will model the effect of v 0. In [20] Hughes has proposed aGreen�s function approach for the solution of the fine-scale problem, while Brezzi [2,4,8] presents a wayto solve it via residual-free bubbles, and Franca et al. [14] propose a two-level finite element approachto perform this task. An equivalence between the Green�s function approach and the use of residual-freebubbles is presented in [5].
Remark 3. It is important to note that the right-hand side of (34) is a function of the residual of the Euler–Lagrange equations for the coarse-scales over the sum of element interiors. It shows that the fine-scaleproblem is in fact driven by the coarse-scale residuals. This is a crucial ingredient of the present multiscalemethod and ensures that the resultant formulation yields a consistent method.
Remark 4. The solution of the fine-scale problem can also be accomplished in the discontinuous Galerkin(DG) framework. This aspect of the fine-scale problem will be presented in a follow up paper.
In order to keep the presentation simple, and to extract the structure of the stability tensor s, we employbubble functions. To crystallize ideas, and without loss of generality, we assume that the fine-scales v 0 andw 0 are represented via bubbles over X 0, i.e.,
where be represents the bubble shape functions over element domains, i = 1, . . .,nsd and b and c representthe coefficients for the fine-scale trial solutions and weighting functions, respectively.
Substituting (35) and (36) in (34) and taking the vectors of constant coefficients out of the integralexpression we get
where I is a nsd · nsd identity matrix, $b is a nsd · 1 vector of gradient of bubble functions. We now recon-struct the fine-scale field over the element domain Xt at time t via recourse to (35)
v0tðxÞ ¼ beðnÞb ¼ beðnÞK�1R. ð41Þ
3.4. Solution of the coarse-scale problem (W)
Let us now consider the coarse-scale part of the weak form W. Exploiting linearity of the solution slot,Eqs. (29) and (30) can be written as
We can combine (50) and (53) to write the HVM form of the incompressible Navier–Stokes equations.Since the resulting equation is expressed entirely in terms of the coarse-scales, for the sake of simplicity thesuperposed bars are dropped.
It is important to note that term number six on the left-hand side has appeared because of the assump-tion of existence of fine-scales in the problem. This term is in fact modeling the numerical subgrid scales inthe problem. Since the method is residual based, therefore the resulting formulation is consistent andaccommodates the exact solution. Another important feature of this formulation is that the structure ofthe stabilization tensor s has appeared naturally via the solution of the fine-scale part of the problem. Fur-thermore, up to the definition of s, a comparison of the HVM method with the GLS and the SUPG meth-ods is straightforward: (a) If we drop the 2mDw from the weighting function slot of the additionalstabilization term, we get the SUPG method. (b) If we change the sign of 2mDw term in the weighting func-tion slot of the additional stabilization term and add the weighting counterpart for the inertial term, we getthe GLS method. In other words GLS and SUPG methods can be recovered from the HVM method, andthat GLS and SUPG methods are subclasses of the HVM method presented here.
Remark 5. The definition of the stability tensor s yields a diagonal matrix only for the case of quadrilateralelements in their rectangular configurations (wherein the interior angles are 90� each). The definition of thestability tensor s that is derived here yields a full matrix for the case of triangles as well as for the distortedquadrilateral elements. It thus activates the cross-coupling effects in the stabilization terms. This cross-coupling effect in the stabilization terms is not present in the GLS or the SUPG methods.
Remark 6. It is important to note that the bubble functions only reside in the definition of the stabilitytensor s. Consequently, the choice of the bubble functions only affects the value of the stability tensor.
Remark 7. As was mentioned in Remark 2, we have assumed that the fine-scale pressure field is zero. Con-sequently, the structure of the stability tensor that is derived via the solution of the fine-scale problem con-tains the basis functions of the fine-scale velocity field alone. When we substitute the modeled fine-scalevelocity field in the coarse-scale problem, the effect is that of enriching the velocity field. This feature ofthe fine-scale problem where we enrich only the velocity field and not the pressure field is attractive fromthe view point of the ratio of constraint count, i.e., the ratio of the velocity to the pressure degrees of free-dom (see [21]).
The numerical examples are broadly divided into two parts. The first part deals with the rate of conver-gence studies where the 2D equal-order velocity–pressure elements have been employed (see Fig. 1). In eachcase optimal quadrature rules are used for numerical integration (see e.g., [21], Chapter 4).
The second part presents various benchmark cases of nonlinear steady and transient flow problems. Forhigh Reynolds number flows, the convergence of the Newton iteration is attained under a good initial guess.Consequently, a continuation process is overlayed the Newton iteration procedure to provide a reasonableinitial guess for the high Reynolds number nonlinear flow problems.
4.1. Rate of convergence study
The first set of numerical simulations present the convergence rates for the formulation. The domainunder consideration is a biunit square, and kinematic viscosity m = 1. The applied body force is given by
In specifying the boundary-value problem, v = 0 is prescribed nodally at the boundary, while the body forceis numerically integrated over the domain. The exact solution to the problem is given by
The rate of convergence study is divided into four parts as follows:
1. The first study investigates the convergence properties of the Stokes part of the operator that is obtainedby setting the skew advection term equal to zero. This study also investigates the attributes of accuratelyevaluating the second order operators that appear in the additional stabilization term in the formulation.
2. The second part of the study presents convergence rates, evaluated on uniform meshes, for the linearizedNavier–Stokes equations.
3. The third part of the study presents convergence rates for the linearized Navier–Stokes equations on dis-torted and graded meshes.
4. The fourth part of the convergence study presents rates for composite meshes wherein different elementstypes of the same order are combined in the computational domain.
The convergence rates for the Stokes operator, employing linear elements, is given in [23]:
Fig. 2.quadri
kevk ¼ Oðh2Þ; krepk ¼ Oðh0:5Þ. ð61Þ
For linear quadrilateral elements, the meshes employed consist of 100, 400, 1600 and 6400 elements. Thelinear triangular element meshes are made by bisecting the quadrilaterals along the major diagonal. For
Convergence rates on uniform meshes for Stokes operator with and without second derivatives: (a) 3-node triangles, (b) 4-nodelaterals, (c) 6-node triangles and (d) 9-node quadrilaterals.
quadratic quadrilateral elements, the meshes employed consisted of 25, 100, 400 and 1600 elements. Again,the quadratic triangular element meshes are obtained by bisection. The element mesh parameter, h, is takento be the edge length of the elements for the quadrilaterals, and the short-edge length for triangles. Thesemeshes were employed in the convergence rate studies presented in Sections 4.1.1 and 4.1.2.
Remark 8. Since the pressure field can be determined up to a constant, so the appropriate norm to checkpressure convergence is the H1-seminorm.
4.1.1. Convergence study for the Stokes part of the formulation
If the convective terms are dropped from Eq. (54), the formulation reduces to the HVM form for theStokes flow problem.
Furthermore, for the Stokes problem, the structure of the stabilization tensor s that emanates from the cor-responding fine-scale problem (49) reduces to
where be is the quadratic bubble function. As was mentioned earlier in Remark 5, Eq. (63) yields a diagonalmatrix only for the case of quadrilateral elements in their rectangular configurations. For the case of trian-gles as well as for the distorted quadrilateral elements s yields a full matrix.
In this section we study the convergence properties of the resulting stabilized Stokes operator for variousequal-order interpolations of the velocity and pressure fields. The stabilization term in (62) contains secondorder operators that require numerical calculation of the second derivatives of the interpolation functions.These second-order operators are also present in the SUPG and the GLS formulations for the Stokes prob-lem. However, most practical applications of the stabilized methods ignore the evaluation of the secondderivatives of the shape functions in the numerical calculations. In the present numerical study we firstinvestigate the effects of ignoring the second derivatives of the interpolation functions on the stabilityand accuracy of the formulation and on the convergence rates of the method. Fig. 2a–d presents the con-vergence rates in the L2-norm for the velocity field and H1-seminorm for the pressure field. The body forceproblem employed herein for the convergence study is obtained by setting a1 = a2 = 0 in (55) and (56). Asshown in Fig. 2a, there is no difference in the two simulations for the 3-node triangles because the second
derivatives of the shape functions are identically zero for the linear triangles. It is interesting to note thatpointwise value of the error decreases with the exact evaluation of the second order terms for the 4-nodequadrilaterals (see Fig. 2b). However, the rate of convergence is the same as for the case wherein the secondderivatives of the shape functions are ignored in the numerical calculations. This situation neverthelesschanges dramatically for the quadratic elements. As shown in Fig. 2d, the exact evaluation of the secondorder terms is crucial to obtaining the theoretical convergence rates for the 9-node quadrilaterals. For thecase wherein the second derivatives in the additional stabilization term are not evaluated exactly, the con-vergence rate becomes suboptimal and the pointwise error also increases. A similar trend is observed for 6-node triangles and is shown in Fig. 2c.
4.1.2. Convergence study for linearized Navier–Stokes equations on uniform meshes
This section presents the convergence study for the HVM form of the linearized Navier–Stokes equa-tions. In this study we are employing the second derivatives of the element shape functions so that wecan exactly calculate the second order terms. The definition of the stabilization tensor s used in this study
(a)
(b)
Fig. 5. (a) Skew meshes of 200 triangles and 100 quadrilaterals and (b) skew meshes of 800 triangles and 400 quadrilaterals.
is given in (49). It is important to note that the use of same bubble function for the trial solution and theweighting function together with the assumptions on the fine-scale velocity field given in (18) and (19) leadsto the cancellation of the skew term in the definition of s. In this work we follow along the lines of Masudand Khurram [32] and use different order interpolation functions for the fine-scale trial solution andweighting function in the skew part of (49). Accordingly, the modified stabilization tensor is defined asfollows:
Fig. 6.(d) 9-n
s ¼ be1
ZXebe1 dX
� � ZXebe2v
c � rbe1 dXI þ mZXejrbe1j
2 dXI þ mZXerbe1 �rbe1 dX
� ��1
; ð64Þ
where be1 is the standard quadratic bubble while be2 represents the bubble for the fine-scale weighting func-tion in the skew part of (64).
Figs. 3a–d and 4a–d present the L2-norm of the velocity andH1-seminorm of the pressure for a1 = a2 = 1and a1 = a2 = 10, respectively. We see optimal rates for velocity field. The rates for the pressure field are lessthan optimal in the norm considered, however, the attained rates are in accordance with the theoreticalpredictions.
Remark 9. In general s is a second order tensor that yields a 2 · 2 matrix in 2D. The advection term onlycontributes to the main diagonal of this matrix, and in the advective limit each of these terms is O(h/jvj).The diffusion term on the other hand contributes to the entire matrix and in the diffusive limit each term isO(h2/m). Consequently in both the limit cases s possesses the right order [24].
4.1.3. Convergence study on distorted meshes
Fig. 5 shows representative distorted and graded meshes used in the convergence study for the quadri-lateral and triangular elements. The degree of distortion has been kept constant amongst the variousmeshes. It is important to note that for the distorted and graded meshes, s turns out to be a full matrixfor all the element types. Fig. 6 presents the convergence rates for various elements. We observe virtually
Fig. 10. Convergence rates for composite meshes for a1 = a2 = 10: (a) linear composite mesh and (b) quadratic composite mesh.
Fig. 11. Schematic diagram of the problem description.
no degradation in the convergence rates for this case as compared to the corresponding rates that were ob-tained on uniform meshes.
4.1.4. Convergence study with composite meshes
Fig. 7 presents composite meshes composed of triangles and quadrilaterals in the same computationaldomain. Since rate of convergence is a function of the order of the interpolation polynomial, therefore thesecomposite meshes should yield the theoretically predicted rates. This study is interesting from a practicalview point where the choice of element types is often based on the geometry of the physical domain as wellas the mesh generator at hand. The qualitative pressure field for the linear elements is shown in Fig. 8a andthe corresponding exact pressure field is presented in Fig. 8b. Once again we attain the theoretically pre-dicted rates for the two advection cases. These convergence rates are presented in Figs. 9 and 10.
Fig. 12. Streamlines for Re = 5200: (a) 20 · 20 4-node element mesh, (b) 40 · 40 4-node element mesh and (c) 80 · 80 4-node elementmesh.
Driven cavity flow serves as a standard benchmark problem for incompressible Navier–Stokes equa-tions. Fig. 11 shows the problem description wherein the flow is driven by applying a unit tangential velo-city on the top surface. The pressure is prescribed to be zero at the center of the cavity. As a result of theprescribed flow at the top of the cavity, a recirculation region is developed that bears a primary vortex inthe middle of the cavity. Depending on the Reynolds number, additional secondary vortices may appear inthe corners of the cavity. Over diffusive numerical methods may not capture the secondary vortices, thusmaking this problem a good test for checking numerical diffusion in the method. Fig. 12 shows the stream-lines for Reynolds number 5200 on three different uniform meshes of 20 · 20, 40 · 40 and 80 · 80 4-nodeelements. In comparison with the two-level finite element methods [14,16], we observe four vortices withuniform meshes of 40 · 40 elements and higher.
Fig. 13. Streamlines superimposed on absolute velocity contours (Re = 10,000): (a) 40 · 40 4-node element mesh, (b) 80 · 80 4-nodeelement mesh and (c) 160 · 160 4-node element mesh.
In the next study the Reynolds number is increased to 10,000. Fig. 13 shows the streamlines superim-posed on the colored velocity distribution for three different uniform meshes of 40 · 40, 80 · 80 and160 · 160 elements. Five resolved vortices are captured, with the exception of the 40 · 40 element mesh,which is too coarse for such a high Reynolds number. The zoomed view of the lower right corner of thecavity is shown in Fig. 14. The pressure contours for the 80 · 80 mesh are shown in Fig. 15.
Following the frequent practice of comparing the velocity profiles with that of Ghia et al. [15] we presentthe line plots for vx along a vertical line passing through the geometric center of the cavity. Fig. 16 showsthe graded mesh of 60 · 60 4-node elements which is employed in the comparative study. Fig. 17 shows thecomparison for Re = 100, 400, 1000, 3200, 5000 and 7500, and a good comparison is attained in the entirerange of flow regimes. The case of Re = 10,000 was solved with both 4-node quadrilaterals and 3-node tri-angles. The triangular mesh was generated by dividing the 4-node element mesh along its main diagonal. It
Fig. 14. Zoomed view of the vortices at the bottom right corner (Re = 10,000): (a) 40 · 40 4-node element mesh, (b) 80 · 80 4-nodeelement mesh and (c) 160 · 160 4-node element mesh.
is important to note that the total number of unknowns are the same in both the cases. Once again a goodengineering accuracy is attained for the two element types as shown in Fig. 18.
4.3. The backward facing step flow
The backward facing step is a benchmark problem and is known to possess a corner singularity. Thegeometry and the boundary conditions are shown in Fig. 19. A fully developed parabolic velocity profileis prescribed at the inlet boundary. The Reynolds number is 150 which is based on the maximum inletvelocity vx(max) = 1 and the height of the inlet. Two different meshes composed of 224 and 896 bilinearquadrilateral elements are employed. Fig. 20a and b shows the contour plots of the pressure field.Fig. 20c shows the streamlines displayed on the colored velocity distribution. The results are in agreementwith the two-level method of Franca and Nesliturk [14].
4.4. Flow past a circular cylinder at low Reynolds number
We next consider the two-dimensional flow of an incompressible fluid past a circular cylinder. The prob-lem description is given in Fig. 21a. A zoomed view of the mesh that consist of 12,000 four node elements,
Fig. 17. Cavity results for a graded mesh of 60 · 60 4-node elements: (a) Re = 100, (b) Re = 400, (c) Re = 1000, (d) Re = 3200,(e) Re = 5000 and (f) Re = 7500.
wherein the nodes are concentrated around the cylinder by employing an exponential function, is presentedin Fig. 21b. The Reynolds number is 40, which is based on the free-stream velocity and the diameter of thecylinder. A steady state solution exists for this Reynolds number. Fig. 22 shows the streamlines that aresuperimposed on the colored pressure field. The wake length is found to be in good agreement with thevalue reported in Pontaza and Reddy [33]. Fig. 23 shows the computed pressure coefficient distributionon the surface of the cylinder and a comparison is made with the experimental data of Grove et al. [17].
4.5. Transient vortex shedding around a circular cylinder
Viscous fluid flow phenomena past circular cylinders involving the unsteady rapidly changing flow pat-terns has been a major area of investigation in computational fluid dynamics. The present test simulates theperiodic wake flow with vortex shedding from the cylinder. The computational domain for this problemscovers an area �6 6 x 6 20 and �6 6 y 6 6, with a unit diameter cylinder having its center point placed atx = 0 and y = 0. For this problem, free stream values of velocity are prescribed (i.e., vx = 1, vy = 0) on theinflow, top and bottom boundaries, while traction free conditions are applied at the outflow. The no slipcondition is prescribed on the surface of the cylinder to account for the adhesion of the viscous fluid tothe surface of the cylinder. The fluid is assumed to be homogeneous and isotropic and the viscosity ofthe fluid m = 0.01. The Reynolds number is 100 which is based on the diameter of the cylinder and the in-flow velocity vx = 1. The backward Euler implicit method is employed for the transient calculations. Timestep Dt = 0.01. The mesh employed in this simulations is shown in Fig. 24.
Fig. 25 shows the stationary streamlines superimposed on the resultant velocity contours at various in-stants in time. First, a symmetric pair of eddies appears behind the cylinder. The length of the wake bubble
Fig. 18. Comparison of linear elements for the cavity problem (Re = 10,000): (a) 7200 3-node triangles and (b) 3600 4-nodequadrilaterals.
Fig. 19. Problem description of the backward facing step.
Fig. 20. The backward facing step problem, Re = 150: (a) pressure contours on 224 4-node element mesh, (b) pressure contours on 8964-node element mesh and (c) zoomed view of streamlines superimposed on velocity contours, 896 4-node element mesh.
Fig. 21. Problem description for stationary flow around cylinder: (a) problem description and (b) mesh.
increases till it becomes unstable, leading to periodic vortex shedding, known as the Karman vortex street.This self-induced vortex shedding produces an unsteady lift force in the lateral direction. The dependentvariables of practical importance in this problem are the lift coefficient CL ¼
R 2p0
r2n d/=pqv2 and the Strou-hal number (St = D/vT), where r2n is the transverse component of the normal stress rn, and v is the free-stream velocity. Fig. 26 shows the temporal development of the lift coefficient. From the plot of lift coef-ficient we obtain the shedding period T = 6.2. It gives a Strouhal number of 0.161 which is in good agree-ment with the value reported in Hauke and Hughes [19].
5. Conclusions
We have presented a new multiscale method for the incompressible Navier–Stokes equations. Thedecomposition of the velocity field into coarse and fine-scales and the introduction of a model for thefine-scale velocity, corrects the lack of stability of the standard Galerkin formulation of the Navier–Stokesequations. Since the proposed method is residual based, it accommodates the exact solution and is thereforea variationally consistent method. From the viewpoint of stabilized methods, the additional terms possess astructure that resembles the structure of stabilization terms in the SUPG and the GLS methods. It is impor-tant to note that in the present method these terms have been derived consistently. In addition, the structureof the stabilization tensor s has also appeared naturally via the solution of the fine-scale problem. The ensu-ing mathematical nesting of the fine-scales into the coarse or resolvable scales also results in higher accuracyon cruder discretizations. A family of 2D triangular and quadrilateral elements has been developed andtheoretically predicted convergence rates on uniform, skewed and composite meshes have been attained.The method is shown to exhibit good stability and accuracy properties when applied to various nonlinearsteady and transient flow problems.
Acknowledgments
Support for this work was provided by the Office of Naval Research grant N00014-02-1-0143. This sup-port is gratefully acknowledged.
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