Abstract This paper proposes a hybrid vertex-centered fi-nite volume/finite element method for solution of the two di-mensional (2D) incompressible Navier–Stokes equations onunstructured grids. An incremental pressure fractional stepmethod is adopted to handle the velocity-pressure coupling.The velocity and the pressure are collocated at the node ofthe vertex-centered control volume which is formed by join-ing the centroid of cells sharing the common vertex. Forthe temporal integration of the momentum equations, an im-plicit second-order scheme is utilized to enhance the com-putational stability and eliminate the time step limit due tothe diffusion term. The momentum equations are discretizedby the vertex-centered finite volume method (FVM) and thepressure Poisson equation is solved by the Galerkin finite el-ement method (FEM). The momentum interpolation is usedto damp out the spurious pressure wiggles. The test case withanalytical solutions demonstrates second-order accuracy ofthe current hybrid scheme in time and space for both veloc-ity and pressure. The classic test cases, the lid-driven cavityflow, the skew cavity flow and the backward-facing step flow,show that numerical results are in good agreement with thepublished benchmark solutions.
This work was supported by the Natural Science Foundation ofChina (11061021), the Program of Higher-level talents of In-ner Mongolia University (SPH-IMU, Z200901004) and the Sci-entific Research Projection of Higher Schools of Inner Mongolia(NJ10016, NJ10006).
W. Gao · H. LiSchool of Mathematical Sciences, Inner Mongolia University,010021 Huhhot, Chinae-mail: [email protected]
R.-X. LiuDepartment of Mathematics,University of Science and Technology of China,230026 Hefei, Chinae-mail: [email protected]
Many computational fluid dynamics (CFD) problems in-volve incompressible flows in complex geometries. Exam-ples include serpentine ducts, curved internals in fossil fuelburners and blood flows in diseased arteries. These motivatethe development of computational schemes on unstructuredand non-orthogonal meshes.
The major difficulty of the numerical solution of the in-compressible flows arises from the velocity-pressure cou-pling of the incompressible equations in the form of theprimitive variables. The SIMPLE method by Patankar andSpalding [1] and the fractional step projection method byChorin [2] and Temann [3] are prevailing in the spectrumof the pressure-correction algorithms in both scientific com-putations and engineering applications. The chief differ-ence between the fractional step method and the SIMPLEmethod is that for the former, the pressure correction Pois-son equation is solved once per time step while for the lat-ter, the momentum and pressure correction equations aresolved several times in each time step. So, the fractionalstep method is more efficient than the SIMPLE method whenwe solve the unsteady incompressible flows [4]. The frac-tional step method by Chorin [2] is of first-order accuracy intime. Afterwards, some improved fractional step methods,possessing the second order temporal accuracy, were pro-posed in published literatures [5–8]. The present study usesthe second-order implicit fractional step method similar toRef. [8].
Under this line, the finite volume method (FVM) is oneof significant spatial discretization approaches on unstruc-tured grids. To performed the FVM discretization, it is nec-essary to define a node and a control volume (CV) surround-ing this node. Three approaches are available: the circum-centered approach, the centroid-centered approach and thevertex-centered approach (see Fig. 1). The circum-centered
A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes 325
CV is employed in few published methods. Its serious dis-advantage is that the circumcenter of each cell may, in gen-eral, not lie within the cell, as shown by the node B inFig. 1a, which often leads to very complex implementationof the solution scheme of the momentum equations in thefinite volume formulation. Boivin et al. [11] and Perron etal. [21] developed the project method in association withthe circum-centered finite volume formulation for the so-lution of the incompressible Navier–Stokes equations. Thecentroid-centered approach is a very popular manner to con-struct the CV on unstructured grids. It defines a node atthe centroid of each cell (see Fig. 1b) and the cell itself istreated as the CV. The line joining two neighboring nodes,in general, will not be orthogonal to the interface shared bytwo adjacent cells. This non-orthogonality may cause dif-ficulties in the finite volume discretizaiton of the diffusionterms of the momentum equations and the pressure Poissonequation on unstructured grids. A sophisticated diamond-cell approach was proposed by Mathur and Murthy [16]to effectively handle this problem. Subsequently, variousfractional step method based the centroid-centered FVMand staggered/non-staggered variable arrangement were pre-sented in many published literatures [12, 23–26]. The vertex-centered approach positions the node at the vertex and a CVis constructed by joining centroids of successive cells sur-rounding the vertex A, as shown in Fig. 1c. Compared toaforementioned two CVs, the vertex-centered approach re-quires less computational memory although its implementa-tion is more complex than that done by other two CVs.
Fig. 1 Typical unstructured meshes: a Circum-centered; bCentroid-centered; c Vertex-centered
Few numerical methods employ the fractional step pro-jection method in association with the vertex-centered FVMin solution of incompressible Navier–Stokes equations be-cause few efficient and mature approaches based on thevertex-centered FVM were proposed to discretize the pres-sure Poisson equation until now. Foy et al. [28] presentedan unstructured vertex-centered FVM fractional step methodfor solution of incompressible Navier–Stokes equations. Hedeveloped a consistent vertex-centered FVM disretization
for the pressure Poisson equation. This method can not bewidely used for its more complex implementation on un-structured vertex-centered grids. In contrast, for the dis-cretization of the Poisson equation on unstructured grids, thefinite element method (FEM) is a better and more efficientmethod than that of the FVM does because it goes beyondthe limitation of the unstructured cell shape. So, a betterapproach for solution of the incompressible Navier–Stokesequations is expected to take advantage of both FVM andFEM, that is to use FVM for discretization of the momen-tum equations and FEM for the pressure Poisson equation.Tu et al. [13] proposed an FVM/FEM method on the unstruc-tured semi-staggered grids. He used a least square approachto damp spurious pressure wiggles. All these cause compleximplementation of codes despite of its good numerical per-formance.
In this paper, a hybrid vertex-centered finite vol-ume/finite element method is proposed for solution of theincompressible Navier–Stokes equations. An incrementalpressure fractional step projection method similar to Ref. [8]is employed to handle the velocity-pressure coupling. Thevertex-centered CV is formed by joining the centroids of suc-cessive cells surrounding a vertex defined as a node of theCV (see Fig. 2). A non-staggered variable arrangement isused in the present study. This means that all primitive vari-ables, u, p, are collocated on the node of the vertex-centeredCV (see Fig. 2). Based on the vertex-centered CV, the FVMis used for the spatial discretization of the momentum equa-tions. For the temporal integration of the momentum equa-tions, an implicit second-order scheme is utilized to enhancethe computational stability and eliminate the time step limitdue to the diffusion term. Three neighboring nodes, A, Band C, shown in Fig. 2, consist of a triangle on which alinear basis function is defined in the finite element frame-work for discretization of the pressure Poisson equation. Bydefining an interface flux (U j) as Zang did in Ref. [10] (seeFig. 2), the momentum interpolation technique proposed byRhie and Chow [9] can be handily implemented on currentvertex-centered CV to eliminate the spurious pressure oscil-lations arising from the non-staggered variable arrangement.
Fig. 2 The geometry of the vertex-centered control volume
326 W. Gao, et al.
2 Governing equations
The Navier–Stokes equations for a 2D viscous incompress-ible laminar flow can be written in a dimensionless form as
∇ · u = 0, (1)∂u
∂t+ (u · ∇)u = −∇p +
1Re∇2u, (2)
on the domain Ω × (0, T ]. Where, Ω denotes the spatialdomain with the boundary ∂Ω, and the time interval is ex-pressed by (0, T ], u and p are the velocity and pressure field,respectively. The entire physical boundary can be classifiedas inlet, wall and outlet, that is, ∂Ω = Γin
⋃Γw⋃
Γo, where
Γin: the inlet boundary, u|Γin = uin,
Γw: the wall boundary, u|Γw = uw,
Γo: the outflow boundary where the fully developed flow is
usually imposed on.
The initial conditions for Eqs. (1) and (2) [14] are
u(x, y, 0) = u0(x, y),
on Ω = Ω⋃∂Ω,
(3)
where u0(x, y) satisfies ∇ · u0(x, y) = 0.
3 Numerical method
The fractional step method is popular for solving incom-pressible Navier–Stokes equations (1) and (2) in terms ofthe primitive variables. Firstly, an approximation to the mo-mentum equations is performed to obtain a tentative velocityfield. Then, an elliptic equation for the pressure or the in-cremental pressure is solved to enforce the incompressibilityconstraint and update the velocity and pressure field.
The momentum equations would be discretized in timeby using several schemes. The explicit Euler scheme is con-ditionally stable and of first-order accuracy. Its counter-part, implicit Euler scheme is strongly A-stable and tendsto suppress oscillations but also of first-order accuracy. TheCrank–Nicolson scheme possesses the second-order accu-racy but is only A-stable. Compared to the Crank–Nicolsonscheme, the second-order accurate implicit time scheme isstrongly A-stable and employed in the temporal discretiza-tion of the momentum equations. So the momentum equa-tions can be disretized in time as follows
β1un+1 + β0u
n + β−1un−1
Δt+ [(u · ∇)u]n+1
= −∇pn+1 +1
Re∇2un+1, (4)
where β1 = 3/2, β0 = −2 and β−1 = 1/2 corresponds to thesecond-order implicit time scheme.
In the predictor step, an analog of the momentum equa-tion (4) is to be solved to yield a tentative velocity field u∗that does not satisfy the incompressibility constraint. In ad-dition, pn, regarded as a better approximation to pn+1, is of-ten used in the predictor step because the pressure field is notavailable at the same time level as the velocity field. So, thefollowing equation is solved in the predictor step
β1u∗ + β0u
n + β−1un−1
Δt+ [(u · ∇)u]∗
= −∇pn +1
Re∇2u∗. (5)
In the projection step, an incremental pressure q is defined asq = pn+1 − pn. According to Eqs. (4) and (5), q satisfies thefollowing equation
β1un+1 − u∗Δt
= −∇q. (6)
Taking the divergence of Eq. (6) and using the incompress-ibility constrain ∇ · un+1 = 0, we obtain a Poisson equationfor the incremental pressure q as
∇2q =β1
Δt∇ · u∗. (7)
Then, the velocity and pressure at the new time level are cor-rected by
un+1 = u∗ − Δtβ1∇q, (8)
pn+1 = pn + q. (9)
In Eqs. (5) and (7), u∗ and q are tentative variables with-out corresponding physical boundary conditions. Consistentartificial boundary conditions proposed by Gresho [14] areemployed on u∗ and q in the present study. Boundary con-ditions for Eqs. (5) and (7) are set as follows
(1) u∗ = uin and∂q
∂n= 0 on the inlet Γin,
(2) u∗ = uw and∂q
∂n= 0 on the wall Γw, and
(3)∂u∗
∂n= 0 and q = 0 on the outlet Γo,
where n means the outer normal vector to the boundary.Note that the artificial pressure Neumann boundary condi-tions q on Γin and Γw will lead to the loss of accuracy for thepressure field. Timmermans et al. [19] presented a correctionto cure this defect. Following their approach, we update thepresent pressure field by
pn+1 = pn + q − 1Re∇ · u∗. (10)
Guermond and Shen [20] proved that Eq. (10) yields about1.5th order accuracy for the pressure in time while the ve-locity is second-order accurate in time. In practice, thecurrent numerical computations and published results in
A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes 327
Refs. [6, 7, 12, 13] presented better performance in the tem-poral accuracy. Now, this full approach is known as the rota-tional incremental fractional step method [13, 19, 20].
3.1 Finite volume discretization for momentum equations
A vertex-centered finite volume method is used for the spa-tial discretization of the momentum equations. A CV isformed by joining centroids of successive triangular ele-ments surrounding a common vertex A or B at which allprimitive variables are positioned (see Fig. 2). Note that theline segment AB, joining two adjacent vertices A and B, ingeneral, will not be orthogonal to the interface cd. Equation(5) is integrated in the CV with the node A. By using thedivergence theorem, their spatial discretizations are yieldedas
ArΔt
(β1u∗ + β0u
n + β−1un−1) +
∫
S[(u · n)u]∗ ds
= −∫
Spnnx ds +
1Re
∫
S∇u∗ · n ds, (11)
ArΔt
(β1v∗ + β0vn + β−1vn−1) +∫
S[(u · n)v]∗ ds
= −∫
Spnny ds +
1Re
∫
S∇v∗ · n ds, (12)
where Ar is the area of the CV, S =⋃
S j is the surface ofthe CV and n = (nx, ny) is the outer normal vector to the cellsurface S .
3.1.1 Discretization of the convection term
The spatial discretization of the convection terms is per-formed for Eqs. (11) and (12). For convenience, the dis-cretization is described in the vectorial form, and the super-scripts “∗” and “n” are omitted. The convection terms can bediscretized as∫
S[(u · n)u] ds ≈
∑
S j
(u · n) ju jΔl j =∑
S j
C(u) j. (13)
The inviscid flux on an interface S j takes the general formof the Godunov scheme widely used as an approximate Rie-mann solver
C(u) j = λ+j u
Lj + λ
−j u
Rj . (14)
The second-order accurate upwind scheme is used to recon-struct the left and the right state of the velocity at the inter-face f
uLj = uA + ∇uA · rA,
uRj = uB + ∇uB · rB,
(15)
and λ±j = (U j ± |U j|)/2 represents the positive and the neg-ative eigenvalues of the inviscid flux Jacobian matrix. TheU j = (u · n) j is called the interface flux as Zang did inRef. [10]. Moreover, the convection velocity u j normal to
the interface f needs to be obtained to evaluate U j definedby the following equation
U j = (u · n) j = u j · n j, (16)
u j = u∗j − Δt(∇p) j, (17)
u∗j =12
(u∗A + u∗B) · n j, (18)
(∇p) j =12
[(∇p)A + (∇p)B] · n j, (19)
where (∇p) j is the pressure gradient at the mid-point of theinterface f. It is noted that u j is calculated by the momentuminterpolation method following from Eqs. (16) to (19) orig-inally proposed by Rhie and Chow in Ref. [9], which canefficiently eliminate the spurious pressure wiggles on collo-cated grids. In addition, ∇ui(i = A, B) in Eq. (15) is thevelocity gradient at the point A or B, and ri(i = A, B) is theunit vector from the point A or B of the upwind control cellto the mid-point j of the interface S j. Here, a second orderpredictor-corrector approach is applied for the reconstructionof the gradient ∇u at the central point (A or B) of the controlcell. The predictor step reads as follows by using the Gaussformula
u j = (1 − α)uA + αuB, (20)
(∇u) =1
Ar
∑
S j
u jn jΔl j, (21)
where α = dA j/(dA j + dB j) is the linear interpolation factorconsidering the irregularity of the control cell. The correctorstep presents
uLj = uA + ∇uA · rA, uR
j = uB + ∇uB · rB, (22)
u j = (1 − α)uLj + αu
Rj , (23)
(∇u) =1
Ar
∑
S j
u jn jΔl j. (24)
At the new time level, the interface flux u∗j · n j is un-known. In order to avoid solving the nonlinear algebraic sys-tem of equations, it is linearized by using a seconder-orderapproximation (2un − un−1) j · n j.
Additionally, the deferred correction method [15] is usedto constitute the algebraic system of equations. At the newtime level, the first term on the right hand side of Eq. (15) iscalculated implicitly whereas the remaining term on the righthand side of Eq. (15) is calculated explicitly by using the val-ues at the previous time levels and treated as the source term.
3.1.2 Discretization of the diffusion term
The spatial discretization of the viscous stress terms will beperformed for Eq. (11), and the analysis for Eq. (12) is simi-lar. For convenience, the superscripts “∗” and “n” are omit-ted. The viscous stress term of Eq. (11) can be discretizedas∫
S(∇u) · n ds =
∑
S j
(∇u) j · n jΔl j =∑
S j
D(u) j. (25)
328 W. Gao, et al.
In order to discretize the first term on the right hand side ofEq. (25), we define the viscous flux at the interface S j as
D(u) j = (∇u) j ·A j,
A j = n jΔl j = (Ax, Ay), (26)
Ax = yd − yc, Ay = −(xd − xc).
In order to express D(u) j by the values of u at the centralpoints A and B, we transform physical coordinates (x, y) tocomputational coordinates (ξ, η) as shown in Fig. 3. Then,the viscous flux can be rewritten by the chain rule
D(u) j = (∇u) j ·A j
= uξ(ξxAx + ξyAy) + uη(ηxAx + ηyAy). (27)
As Mathur and Murthy did in Ref. [16], the viscous flux atthe interface S j can be expressed as
(∇u ·A) j =(uA − uB)Δξ
A j ·A j
A j · rξ
+ (∇u)∗ ·(
rξA j ·A j
Δη− rξA j ·A j
rξ ·A j
)
, (28)
where Δξ is the distance between two vertices A and B, Δη isthe length of the interface S j and (∇u)∗ = (1−α)∇uA+α∇uB.The expression consists of two parts: the first part on theright hand side of Eq. (28) represents the primary-diffusionterm and is equivalent to a second-order central differencediscretization; the second part represents the cross-diffusionterm which vanishes if rξ · rη is zero (e.g. on an orthogonalmesh).
Fig. 3 The local geometry of the diamond cell formed by two ad-jacent vertices A and B and two centroids c and d
Additionally, the deferred correction method [15] is usedagain to handle the diffusion terms for constituting the alge-braic system of equations. The first term on the right handside of Eq. (28) is calculated implicitly. The second terms onthe right hand side of Eq. (28) are calculated explicitly andtreated as source terms.
3.2 Finite element discretization for the Poisson equation
Within every time step, a Poisson equation needs to be solvedfor the incremental pressure before updating the velocity andthe pressure fields. The Poisson equation is a typical ellipticequation. Its resulting boundary condition is the Neumanntype or the mixed type according to the physical boundaryconditions. So, compared to the FVM, a better manner todiscretize the elliptic equation is the vertex-based GalerkinFEM especially on unstructured triangular meshes. The La-grangian linear basis function on every vertex is denoted byNi(x, y), i = 1, 2, 3. The nodal incremental pressure q can beexpressed on every triangular element by
q =3∑
i=1
qiNi(x, y). (29)
The relevant q is linear on every triangular element and ofC0 continuity on every element border. The variational formof Eq. (7) is obtained by multiplying it with a test functionφ(x, y) and integrating on the whole domain Ω by Green for-mula
Left hand side is
−∫
Ω
(∇q · ∇φ) dΩ +∫
∂Ω
(∇q · n)φ ds,
Right hand side is
− 1Δt
∫
Ω
(u∗ · ∇φ) dΩ +1Δt
∫
∂Ω
(u∗ · n)φ ds,
(30)
Taking the boundary conditions into account, we obtain∫
Ω
(∇q · ∇φ) dΩ
=1Δt
[∫
Ω
(u∗ · ∇φ) dΩ −∫
Γin⋃
Γw
(u∗ · n)φ ds
]
. (31)
The discretization form of Eq. (31) reads
Ne∑
n=1
∫
en
∇qh · ∇φh dΩ =1Δt
Ne∑
n=1
∫
en
u∗ · ∇φh dΩ
− 1Δt
Ne∑
n=1
∫
γn
(u∗ · n)φn ds, (32)
where en denotes a triangular element, and Ne is the num-ber of the triangular element. The
⋃Nen=1 en = Ωh is the dis-
cretization of the domain Ω. The⋃Ne
n=1 γn = ∂Ωh is the dis-cretization of the boundary ∂Ω. Note that the line integral,the second term on the right hand side of Eq. (32), equalszero when en is the interior element.
Then the incremental pressure gradient on every vertexcan be obtained by
∇q ≈ 1Ar
∫
S
qn ds. (33)
Note that a second order approximation of the gradient inEq. (33) can be obtained by a predictor-corrector approach
A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes 329
similar to the procedure from Eq. (20) to (24). These se-cure no loss of the accuracy in updating the velocity by usingEq. (8).
3.3 Solution of the algebraic equations
After the aforementioned FVM and FEM discretizations arepreformed, the following algebraic equations are obtained
AΦ = S, (34)
where A is the coefficients matrix, Φ denotes unknowns andS is the source term. Due to the present unstructured tri-angular grids, the resulting coefficient matrix is sparse andunbounded. In order to take advantage of the sparse charac-teristics in solving the algebraic equations, iteration methodsare usually employed. An effective and popular algorithmsuitable for the unstructured-grid calculations is the gener-alized minimal residual method. It is employed in solutionsfor both the momentum equations and the pressure Poissonequation. The convergence criterion is that the L2 norm ofthe residuals over all control cells is less than the prescribedvalue, usually 10−5,
R = AΦ − S,‖R‖L2 < 10−5,
(35)
where R denotes the residual.
4 Numerical results and discussion
4.1 Decaying vortices
The Taylor problem is the first test case to validate the tem-poral and spatial accuracy of the current solution method.This problem corresponds to periodic and counter-rotatingvortices whose strength decays in time at a rate determinedby the viscosity. The analytical solution of the Taylor prob-lem is given by [12, 26, 27]
u(x, y, t) = sin(πx) cos(πy) exp(−2π2t/Re), (36)
v(x, y, t) = − cos(πx) sin(πy) exp(−2π2t/Re), (37)
p(x, y, t) =14
[cos(2πx) + cos(2πy)] exp(−4π2t/Re), (38)
on a square domain Ω = [0, 1] × [0, 1]. The initial condi-tions on the velocity and the pressure are given at t = 0. Thevelocity boundary conditions in time is provided from theanalytical solutions. Present calculations are performed witha Reynolds number fixed at Re = 10. The Reynolds numberis defined as Re = UL/ν, where U is the initial maximumvelocity and L is the domain length scale.
The following error norms for the velocity and the pres-sure, respectively, are defined to demonstrate the differences
between the numerical solution and the exact solution
‖ev‖ =
√√√√ Ne∑
i=1[(uh − ue)Ae]2 + [(vh − ve)Ae]2
Ne,
‖ep‖ =
√√√√ Ne∑
i=1[(ph − pe)Ae]2
Ne,
(39)
where Ne is the number of the vertex-centered CVs and Ae
is the area of every CV. The subscripts “h” and “e” denotethe numerical and exact solutions, respectively. The order ofconvergence is computed by
order =ln(‖e‖τ+1/‖e‖τ)
ln 0.5, (40)
where the subscript “τ” denotes the spatial or temporal meshrefinement level.
Compared solutions are obtained on three grids with themesh size h = 0.05, 0.025 and 0.0125, respectively, usedin the Easymesh. The coarsest grid generated by Easymeshwith the mesh size h = 0.05 is shown in Fig. 5. The accu-racies in time and space are investigated, respectively. Thevariations of the error norms of the velocity and pressure areillustrated in Fig. 6 by the varying mesh size while keep-ing the time step constant, and therefore the slope shown inFig. 6 denotes the spatial accuracy. Note that a small constantvalue of the time step (Δt = 0.000 1) is used to guaranteethat the temporal error is small in comparison with the spa-tial discretization error when testing the spatial convergence.Similarly, the temporal accuracy is investigated by varyingthe time step but keeping the mesh size constant. Here, thefinest grid with the mesh size h = 0.012 5 is used to mini-mize the spatial error. Figure 7 shows that the present timeintegration scheme is indeed second-order accurate.
Fig. 4 The geometry of the FEM triangle formed by three adjacentvertices A, B and C
Fig. 5 The coarsest grid with the mesh size h = 0.05 generated byEasymesh
330 W. Gao, et al.
Fig. 6 Spatial convergence rates for the velocity and pressure
Fig. 7 Temporal convergence rates for the velocity and pressure
4.2 The lid-driven cavity flow
The lid-driven cavity is a well known benchmark problemfor the incompressible flow solvers. It is also employedto validate the current hybrid solver. The Reynolds num-ber is defined by Re = UL/ν with U being the velocity ofthe top moving lid and L being the lid length. We use thepresent solution method to simulate the Newtonian flow withRe = 1 000 on three unstructured triangular grids. The coars-est grid, grid a, consists of 930 vertices by using the meshsize h = 0.05 in Easymesh. The finer grid, grid b, consists of3 698 vertices by using h = 0.025, the finest grid c has 14 818vertices by using h = 0.012 5. Figures 8 and 9 illustrate thecomputed streamlines and the pressure field on the grid c.The smooth pressure field can be observed without spuriousoscillations. The present results are also compared with themost referenced benchmark data by Ghia et al. [17]. Figures10 and 11 show the u-velocity along the vertical centerlineand the v-velocity along the horizontal centerline, which arecompared with Ghia’s results, respectively. With increasinggrid refinement, it is seen that the current results match the
benchmark solution better. And the numerical solution com-puted on the finest grid, grid c, agrees well with the referencesolution.
Fig. 8 Streamlines of the lid-driven cavity at Re = 1 000
Fig. 9 Pressure contours of the lid-driven cavity at Re = 1 000
Fig. 10 Comparisons of the u-velocity along the vertical centerlinebetween the present and benchmark solutions
A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes 331
Fig. 11 Comparisons of the v-velocity along the horizontal center-line between the present and benchmark solutions
4.3 The skew cavity flow
Skew cavity laminar flows were set up by Demirdzic etal. [22] for non-orthogonal grids as benchmark cases. Itsgeometry is obtained by skewing the unit square to the onewith an inclination angle β as shown in Fig. 12. Zero veloc-ity components are imposed on all walls except for the upperlid which slides with an unit tangential velocity. Followingthe benchmark solutions by Demirdzic et al. [22], the presentcalculations are fulfilled at two Reynolds numbers of 100 and1 000 and two inclination angles of 30◦ and 45◦. Generatedby using the mesh size h = 0.012 5 in Easymesh, the unstruc-tured grid consists of 7 523 vertices for β = 30◦ and 10 784vertices for β = 45◦. Streamlines at Re = 1 000 are shown inFigs. 13 and 14 for the flows in the skew cavity with β = 30◦and 45◦, respectively. These results are quite similar to thoseof Demirdzic et al. [22]. Following Ref. [22], we find thatthe y coordinate for u velocity and the x coordinate for v ve-locity have been normalized along the two centerlines CL1and CL2, respectively. The Cartesian velocity u and v alongthe centerlines CL1 and CL2 are compared in Figs. 15a and15b with the benchmark solutions [22] for Re = 100 and1 000. Figures 16a and 16b show good agreement betweenthe results of Ref. [22] and present results.
Fig. 12 The geometry of the skew cavity
Fig. 13 The streamlines at Re = 1 000 for the skew cavity withβ = 30◦
Fig. 14 The streamlines at Re = 1 000 for the skew cavity withβ = 45◦
Fig. 15 Comparisons of velocity components along two centerlinesof the skew cavity with β = 30◦ a u-velocity, b v-velocity
332 W. Gao, et al.
Fig. 16 Comparisons of velocities components along two center-lines of the skew cavity with β = 45◦
4.4 The backward-facing step flow
The flow behind a backward-facing step in a channel is alsoa widely used benchmark problem to examine the accuracyof numerical methods for incompressible flows. The compu-tational geometry is plotted in Fig. 17. The expansion rate isfixed at 1:2 and the length of the channel is set to 30 h, whereh is the step height. A parabolic inflow velocity is prescribedat the inlet, no-slip and no-penetration conditions for veloc-ity at the walls and a fully developed outflow velocity at theoutlet. The Reynolds number is defined as Re = U(2h)/ν,where U is the average inflow velocity. Calculations are per-formed at Re = 200, 400, 600 and 800, respectively.
Fig. 17 The flow geometry for the backward-facing step
Figure 18 shows the calculated reattachment length asa function of the Reynolds number, which is compared
with some previous results of Armaly et al. [30], Kim andMoin [6] and Barton [31]. The current results agree well withthe reported results of Kim and Moin [6] and Barton [31] forall selected Reynolds numbers. However, a difference oc-curs at Re > 400 between the numerical (present, Refs. [6]and [31]) and experimental (Ref. [30]) results. The three-dimensionality of the flow in experiments may lead to thisdifference as Armaly et al. [30] pointed out. Streamlines aredemonstrated for four Reynolds numbers in Figs. 19a–19c. Itis seen that a secondary separation zone exists on the upperwall at Re = 600 and 800.
Fig. 18 Reattachment length versus the Reynolds number
Fig. 19 Streamlines of the backward-facing step flow for selectedReynolds numbers. a Re = 200; b Re = 400; c Re = 600; dRe = 800
4.5 Comparison with some classical methods
To demonstrate the present computational performance, wecompared the present hybrid solver with other two classi-cal methods, an FVM/FVM solver by Kim et al. [12] andan FVM/FEM solver by Tu et al. [13]. The unstructuredmesh with 9 942 centroid-centered control cells is used forthe next computation. The first test case is the square cav-ity flow at Re = 1 000. The u-velocities along the vertical
A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes 333
centerline and the v-velocities along the horizontal center-line computed by three methods are illustrated in Fig. 20. Itcan be seen that the results by the present hybrid methodare similar to the results by the FVM/FEM method. Theresults by the present method and Tu’s method agree wellwith the reference solution of Ghia [17]. In contrast, the re-sults by the FVM/FVM method [12] show significant differ-ence with the reference solution and the results by other twomethods. Secondly, for the skew cavity flows, the minimumsof u-velocities along the vertical centerline are reported forthe two cases with the inclination angle β = 30◦, 45◦ atRe = 1 000 in Table 1. The minimums of u-velocities alongthe vertical centerline are compared with the benchmark so-lution by Demirdzic et al. [22]. Comparisons show that thepresent method and the Tu’s method can obtain satisfac-tory results in good agreement with the reference solutionwhereas the result by the FVM/FVM method still performsconsiderable deviation from the reference solution. It maybe concluded that the hybrid solvers based on the FVM andFEM possess better computational performance than thatdone by the FVM/FVM solvers. Additionally, for the squarecavity flow at Re = 1 000, comparisons of the CPU-times bythese three methods are listed in Table 2 to show their com-putational efficiency. The CPU-times are obtained after 200time steps and on a coarser mesh with 3 686 triangular con-trol cells by using a PC with a 2.13 GHz Intel Core i3 CPU.For the FVM/FVM method, the solution of the momentumequations spends the CPU-time as 3 times as the solution ofthe pressure Poisson equation. Meanwhile, for the presentmethod and Tu’s FVM/FEM method, the former spends theCPU-time as 10 times as the latter which shows that the so-lution of the Poisson equation takes most of the total solutiontime. So, a better acceleration technique of the Poisson equa-tion coupled into the solver will be our future work. Notethat the Tu’s FVM/FEM method takes more CPU-time thanthe present FVM/FEM method due mainly to its least squareprocedure to suppress spurious pressure oscillations.
Fig. 20 Comparisons of the u (v)-velocity along the vertical (hori-zontal) centerline between the present and other two methods
Table 1 Skew cavity with 30◦ and 45◦ at Re=1000. Comparisonsof the minimum of the u-velocity along the vertical centerline bydifferent methods
β = 30◦,Re = 1 000Methods
y umin(y) Difference
Demirdzic [22] 0.789 –1.973 —
Present — –1.926 2.4%
FVM/FEM [13] — –1.939 1.7%
FVM/FVM[12] — –1.713 13.1%
β = 45◦,Re = 1 000Methods
y umin(y) Difference
Demirdzic [22] 0.793 –1.495 —
Present — –1.462 2.2%
FVM/FEM [13] — –1.475 1.3%
FVM/FVM [12] — –1.316 11.9%
Table 2. The CPU-times by three methods for the square cavityflow at Re = 1 000
CPU-time
Methods Momentumequations
Poissonequations
Total
Present 3.2 32.9 36.1
FVM/FEM [13] 3.2 36.2 39.4
FVM/FVM [12] 3.0 9.4 12.4
5 Conclusion
We present a detailed description of a novel hybrid vertex-centered FVM/FEM for solution of the two dimensionalincompressible Navier–Stokes equations on unstructuredgrids. An incremental pressure fractional step method isadopted to handle the velocity-pressure coupling. The veloc-ity and the pressure are collocated at the node of the vertex-centered CV which is formed by joining the centroid of cellssharing the common vertex. The momentum equations arediscretized by the vertex-centered FVM and the pressurePoisson equation is solved by the Galerkin FEM. The mo-mentum interpolation is used to damp out the spurious pres-sure wiggles.
The test case with analytical solutions demonstrates thesecond-order accuracy of the current hybrid scheme in timeand space for both velocity and pressure. The classic testcases, the lid-driven cavity flow, the skew cavity flow andthe backward-facing step flow, show that numerical resultsare in good agreement with the published benchmark solu-tions.
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