Vorticity Dynamics and Numerical Resolution of Navier-Stokes Equations

Home , Vorticity, Vorticity equation

Mathematical Modelling and Numerical Analysis ESAIM: M2AN Modélisation Mathématique et Analyse Numérique Vol. 35, No 2, 2001, pp. 313–330

∗†

VORTICITY DYNAMICS AND NUMERICAL RESOLUTION OF NAVIER-STOKES EQUATIONS

Matania Ben-Artzi1,∗, Dalia Fishelov2 and Shlomo Trachtenberg 3,†

Abstract. We present a new methodology for the numerical resolution of the hydrodynamics of incompressible viscid newtonian fluids. It is based on the Navier-Stokes equations and we refer to it as the vorticity projection method. The method is robust enough to handle complex and convoluted configurations typical to the motion of biological structures in viscous fluids. Although the method is applicable to three dimensions, we address here in detail only the two dimensional case. We provide numerical data for some test cases to which we apply the computational scheme. Mathematics Subject Classification. 35Q30, 65M06, 76D17.

Received: September 20, 2000. Revised: December 24, 2000.

1. Introduction We present a new methodology for the numerical resolution of the dynamics of incompressible viscid New- tonian fluids. We focus here on the two-dimensional case. The method is applicable to the three-dimensional case as well, as we shall show in a forthcoming paper. The underlying equations in this study are the Navier-Stokes equations [8, 23], which we recall below. In order to emphasize the new aspects in our approach, we restrict ourselves to the simplest case, namely, flow in a closed vessel, subject to “no slip” (zero velocity) boundary conditions. However, in some of our numerical examples, we use more general boundary conditions. An important goal in our study is the calculation of motion of complex bodies in viscous fluids in a biological setting (see below). However, from the mathematical point-of-view, it means that the method should be sufficiently robust to handle rather convoluted or complex configurations. So let Ω ⊆ R2 be a bounded domain with smooth boundary Γ = ∂Ω. Let u(x,t)=(u1(x,t),u2(x,t)), x =(x1,x2) ∈ Ω be the velocity field for t ≥ 0. The Navier-Stokes equations for incompressible viscid flow in Ω are

∂tu +(u ·∇)u = −∇p + ν∆u, ∇·u = 0 (1.1)

∗Partially supported by the Division for Research Funds, Ministry of Science, Israel. †Partially supported by the US-Israel Binational Science Foundation and the Israel Academy of Sciences. Keywords and phrases. Navier-Stokes equations, vorticity-streamfunction, numerical algorithm, vorticity boundary conditions. 1 Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel. e-mail: [email protected] 2 School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel. e-mail: [email protected] 3 Department of Membrane and Ultrastructure Research, The Hebrew University-Hadassah Medical School, P.O. Box 12271, Jerusalem 91120, Israel. e-mail: [email protected] c EDP Sciences, SMAI 2001 314 M. BEN-ARTZI ET AL. where p = p(x,t) is the pressure and ν>0 is the (constant) viscosity coeﬃcient. Equations (1.1) are supple- mented by giving the initial condition for velocity

u(x, 0) = u0(x), x ∈ Ω (1.2) and a “no slip” boundary condition

u(x,t)=0, x ∈ Γandallt ≥ 0. (1.3)

It has become common practice to refer to (u,p) as “primitive variables”. The most extensive theory concerning existence and uniqueness of solutions is formulated using these variables. ThisR theory, originated by Leray [24] | |2 ∞ and Ladyzhenskaya [22] assumed (at least) that the total energy is finite (i.e., Ω u(x,t) dx < ). It proceeds to show the existence and uniqueness of a “continuously evolving” velocity field in this “energy space”. We refer the reader to the books [22, 32] for comprehensive treatments of this theory, labeled as the “Galerkin Method” and to [32] for its numerical implementation. Another approach to the numerical resolution of equations (1.1) is the so-called “projection methodology”. Here the first equation is discretized to finite time intervals and the velocity field u(x,t) is updated to the next time level as u(x,t+∆t). This field is now projected onto the subspace of divergence-free fields, so as to satisfy the (discretized version of) the incompressibility condition ∇ · u = 0. We refer to [2,6,18,31] for detailed accounts. One major drawback of the “primitive variables” approach lies in the difficulties encountered in the study of “vortical flows”, i.e., flows involving vortices, vortex lines or areas of highly concentrated vorticity. From the mathematical point of view, the energy is then infinite [8], beyond the above-mentioned theoretical framework. From the numerical point of view, it becomes necessary to follow directly the evolution of the vorticity. Needless to say, such vortical flows represent a large body of common phenomena in various scientific studies (meteorology, aeronautical science...). In particular, we aim at the hydrodynamic study of biological phenomena and structures spanning a wide range of dimensions, velocities, viscosities, and Reynolds numbers. Biological organisms, single or multicellular, are self-propelled and have dynamic, complex, and convoluted surfaces (boundaries). Propulsion (swimming) is carried out by whole body undulations or by means of spe- cialized propellers capable of changing their geometries and physical properties. The analytical methods we propose here should be suitable to handle and analyze the hydrodynamics and microhydrodynamics of complex biological structures in their natural environments. The “vorticity formulation”, which lies at the basis of our present study, has been very extensively used in computational fluid dynamics, generating a panoply of so called “vortex methods”. We refer the reader to the book [19] and the references cited therein. In the next section, we shall briefly touch further on some aspects associated with these methods. Our approach is based on the “vorticity-streamfunction” formulation of the equations. It is closely related to the work of Dean-Glowinski-Pironneau [9], Quartapelle and coworkers [29, 30] and W.E and J.-G. Liu [11–13]. See also [26,33]. The underlying philosophy of this method is to approximate the vorticity field by “sufficiently smooth” objects (see [3,4] for works justifying the “smoothing effect” in the vorticity equation). We then track its time evolution in terms of its associated streamfunction. Here the biharmonic operator plays a crucial role in the projection of the updated vorticity back onto the subspace where the “vorticity dynamics” takes place. Indeed, this is the vorticity analog of the divergence-free velocity fields. An important corollary of this approach is that we are able to avoid completely the difficulty of determining the vorticity on the boundary Γ . This means that “vorticity generation on the boundary” (a major problem encountered by standard methods) is automatically incorporated here. In fact, at each time step the no-slip condition is implemented via a modification of the whole vorticity field and not just its boundary values (as is the practice of all existing methods) . In Section 2 we recall the vorticity-streamfunction formulation. We then introduce in detail our discretized scheme and the way it handles the boundary conditions. We compare our treatment with that suggested by other authors using the same formulation. Then in Section 3 we give some numerical examples. In the first VORTICITY DYNAMICS AND NUMERICAL RESOLUTION OF NAVIER-STOKES EQUATIONS 315 two examples we measure the convergence rate of the scheme. Then, the time evolution of two vortices with opposite signs is represented. Numerical results are shown for a driven cavity, a double-driven cavity, a driven cavity with a solid small square inside the domain and a uniform flow over a square cylinder. Finally, we treat a backward facing step, where a recirculation zone behind the step is clearly captured.

2. The vorticity-projection method The vorticity function, which is scalar in the two-dimensional case, is deﬁned by

2 1 ξ(x,t)=∇×u = ∂x1 u − ∂x2 u , x ∈ Ω. (2.1)

Taking the curl of the ﬁrst equation in (1.1) we get

∂tξ +(u ·∇)ξ = ν∆ξ, ξ0(x, 0) = ∇×u0. (2.2)

The velocity ﬁeld u is obtained from ξ by using (2.1) in conjunction with ∇·u = 0. This can be done via the “streamfunction formulation” [23] as follows. Due to ∇·u =0,wehaveafunctionψ(x,t) such that ∂ψ ∂ψ u(x,t)=∇⊥ψ = − , , x ∈ Ω,t≥ 0 (2.3) ∂x2 ∂x1 so that invoking (2.1),

∆ψ = ξ (2.4)

Thus, equations (2.2)-(2.4), along with the relation (2.3), serve as the “vorticity-streamfunction” formulation of the problem. We note ﬁrst that by integration of (2.1), the “no-slip” condition (1.3) implies Z ξ(x,t)dx =0, all t ≥ 0 (2.5) Ω

• ∈ 2 2 Thus, for all t, ξ( ,t) L0(Ω) , the L functions of mean-value-zero. Furthermore, the relation (2.3) and the “no slip” condition yield

∂ ψ(x,t)= ψ(x,t)=0, x ∈ Γ,t≥ 0 (2.6) ∂n

∂ ( ∂n is the normal derivative). Note that (2.6) follows since ψ = const. on the boundary and is only determined up to an additive constant. Using standard Hilbert space terminology [32], equation (2.6) can be restated as

• ∈ 2 ψ( ,t) H0 (Ω) (2.7)

2 ∞ 2 where H0 (Ω) is the closure of C0 (Ω) in the H norm. Finally, from (2.4), we infer that • ∈ 2 2 ξ( ,t) ∆(H0 (Ω)) = the image of H0 (Ω) under ∆. (2.8)

2 ⊆ 2 In particular, it is easily verified that ∆(H0 (Ω)) L0(Ω), so that (2.8) implies (2.5). Moreover, it can be 2 2 shown [3] that ∆(H0 (Ω)) is a closed subspace of L0(Ω), so that 2 2 ⊕ L0(Ω) = ∆(H0 (Ω)) K0(Ω) (2.9) 316 M. BEN-ARTZI ET AL. where ⊕ denotes “direct sum” in Hilbert space and K0(Ω) is the subspace of harmonic functions of mean-value- zero. 2 We note that ∆(H0 (Ω)) is the subspace of “vorticity dynamics” (associated with the “no slip” boundary condition) and that equation (2.8) is the basis of our method. When solving the system (2.2)-(2.4) we note that the streamfunction ψ is subject to two (Dirichlet + Neu- mann) boundary conditions, thus rendering (2.4) as an overdetermined equation (for a given ξ). On the other hand, no (explicit) boundary conditions are provided for the vorticity, so that (2.2) is underdetermined. Thus, “the vorticity-streamfunction system is inextricably coupled” [17] p. 429. From both the mathematical and physical points of view the correct assignment of the vorticity on the boundary Γ is crucial in obtaining accu- rate simulations of the flow. In particular, these boundary values determine the shape of the “boundary layer”. In the context of “vortex methods” the most common treatment is that of deriving suitable boundary values for ξ based on a simplified model, such as the “Prandtl model”. We refer to [7,8,14,19] for further details. In the context of the present vorticity-streamfunction formulation, the problem of “vorticity boundary conditions” has been studied by Anderson [1], Quartapelle et al. [10,29,30] and E and Liu [11–13]. Roughly speaking, the basic idea is to derive linear (“orthogonality”) conditions that must be satisfied by the vorticity, as is seen from (2.9). This is a set of linear equations corresponding to the discrete analog of the subspace K0(Ω) of harmonic functions. However, “unfortunately, this full system of equations has a rather cumbersome profile since the equations expressing the integral conditions have almost all coefficients different from zero” [10] p. 878. This is remedied and simplified in the method proposed by Dean, Glowinski and Pironneau [9] by splitting equations (2.2)-(2.4) into two separate Poisson equations. Another “localized” approach to the vorticity boundary conditions was presented by E and Liu [11–13]. To describe our approach for the discretization of (2.2)-(2.4), assume that a first approximation ξ(1)(x,t+∆t) has been obtained from ξ(x,t) by using equation (2.2) (in discretized, explicit form). Then, we find the associated streamfunction ψ(x,t+∆t) by applying the Laplacian to (2.4),

2 (1) ∈ 2 ∆ ψ =∆ξ ,ψ H0 (Ω). (2.10)

∈ 2 2 Indeed, by (2.7), we seek ψ H0 , and standard elliptic theory gives that ∆ is the natural (Dirichlet) operator for this boundary-value problem. Finally, with ψ deﬁned by (2.10), we redeﬁne

ξ(x,t+∆t)=∆ψ(x,t+∆t) (2.11) which is then used as the final value of the vorticity field at time t +∆t. Observe that this definition serves (1) 2 indeed as the projection of the first approximation ξ (x,t+∆t) on the subspace ∆(H0 (Ω)) . Here, not only the boundary values of ξ(1) are adjusted to conform with the “no slip” condition, but the whole vorticity field is suitably modified. In practice, our algorithm consists of the following discretization of equations (2.2), (2.10) and (2.11). To simplify the exposition, we assume that Ω = [a, b] × [c, d]. We lay out a uniform grid

xi = a + i∆x, yj = c + j∆y, tn = n∆t, for i =0, ··· ,M, j =0, ··· ,K, n=0, ··· ,N,

b−a d−c where ∆x = M , ∆y = K and ∆t is determined by the stability considerations. VORTICITY DYNAMICS AND NUMERICAL RESOLUTION OF NAVIER-STOKES EQUATIONS 317

n n Denote by ξij and ψij approximations for ξ and ψ, respectively, at (xi,yj,tn). For any discrete function fij , deﬁne

f − f − δ f = i+1,j i 1,j x ij 2∆x f − f − δ f = i,j+1 i,j 1 y ij 2∆y f − − 2f + f δ2f = δ (δ f )= i 1,j ij i+1,j x ij x x ij (∆x)2 f − − 2f + f δ2f = δ (δ f )= i,j 1 ij i,j+1 · y ij y y ij (∆y)2

n n Given the discretized vorticity and streamfunction, ξij and ψij ,attimet = n∆t , we advance then to t = (n +1)∆t by the following algorithm: (a) In the ﬁrst step, ξ is updated by a discretization of equation (2.2), i.e.,

¯n+1 n − {− n · n n · n } { 2 n 2 n } ξij = ξij ∆t δyψij δxξij + δxψij δyξij + ν∆t δxξij + δyξij (2.12) for all interior points i =1, ··· ,M − 1,j=1, ··· ,K − 1 . On the boundary, i =0,M or j =0,N, ξ is left ¯n+1 n unchanged, i.e., ξij = ξij . n n iα∆x iβ∆y Letting ξij = G e e , we have for the linear version of (2.12), that 4ν∆t α∆x 4ν∆t β∆y 2 ∆t ∆t 2 |G|2 = 1 − sin2 − sin2 + u sin(α∆x)+v sin(β∆y) , (∆x)2 2 (∆y)2 2 ∆x ∆y where u and v are (frozen) constants. If ν ≤ 1 the stability condition |G|≤1+O(∆t) is clearly satisﬁed when 1 1 1 ∆t + ≤ · (∆x)2 (∆y)2 2

(Observe that, in fact, if ν<<1, we can replace the above condition by 1 1 1 ∆t + ≤ min( ,C), (∆x)2 (∆y)2 2ν where C>0 can be chosen rather large). We note here that with some modiﬁcations, the time integration can be converted to an implicit Crank- Nicholson scheme. Such a scheme will be presented later on in this section. (b) In the second step of the algorithm, we construct ψ from ξ by discretizing (2.10), i.e.,by

2 2 n+1 2 2 n+1 2 2 n+1 2 ¯n+1 2 ¯n+1 δxδxψi,j +2δxδyψi,j + δyδyψi,j = δxξi,j + δyξi,j (2.13) for all interior points i =2, ··· ,M−2,j=2, ··· ,K−2, i.e., for all points excluding boundary points and near- boundary points. On the boundary points i =0,M or j =0,K, ψ = 0 is applied, and on near-boundary points − − ∂ψ ∂ψ i =1,M 1orj =1,K 1, ∂n is approximately imposed. For example, for i =1ψ(1,j)=ψ(0,j)+∆x ∂x (0,j), ∂ψ where ∂x (0,j) is known from the boundary condition (2.6). Note that the left hand side of (2.13) can be ··· − ··· − n ··· ··· computed for i =2, ,M 2,j=2, ,K 2 from the values of ψi,j at i =0, ,M, j =0, ,K,and ··· − ··· − ¯n+1 the right hand side of (2.13) can be computed for i =2, ,M 2,j=2, ,K 2 from the values of ξi,j at 318 M. BEN-ARTZI ET AL. i =1, ··· ,M− 1,j=1, ··· ,K− 1, i.e., vorticity values at interior points (excluding boundary points i =0,M or j =0,K). The resulting set of equations is of the form AΨ=B , where Ψ is an unknown vector of the discrete streamfunction values at interior points and A is a symmetric banded matrix. The latter may be solved by a Cholesky factorization of A into LLT ,whereL is a banded lower triangular matrix. In [5] Bjorstad improved the latter by decomposing A into matrices that represent discrete operators in each spatial direction. The resulting algorithm has a complexity of O(MK). (c) Finally, we redeﬁne ξ at t =(n +1)∆t by a discrete version of the projection described in equation (2.11), i.e.,by

n+1 2 n+1 2 n+1 ξi,j = δxψi,j + δyψi,j (2.14) for all points i =0, ··· ,M, j =0, ··· ,K . The boundary values are determined by one-sided diﬀerentiations. Thus, equations (2.12)-(2.14) determine the discrete values of ξ and ψ at t =(n +1)∆t. One can also construct an implicit second order scheme (Crank-Nicholson) as follows.

ξ¯n+1 − ξn ν ij ij = −{−δ ψn+1/2 · δ ξn+1/2 + δ ψn+1/2 · δ ξn+1/2} + {δ2ξn + δ2ξn + δ2ξn+1 + δ2ξn+1} (2.15) ∆t y ij x ij x ij y ij 2 x ij y ij x ij y ij

n+1/2 Here ξij is obtained by Taylor expansion of ξ, using the diﬀerential equation (2.2), i.e.,

∆t ξ(x ,y ,t+∆t/2) = ξ(x ,y ,t)+ ξ (x ,y ,t)+O(∆t2) i j i j 2 t i j ∆t = ξ(x ,y ,t)+ {−(u(x ,y ,t) ·∇)ξ(x ,y ,t)+ν∆ξ(x ,y ,t)} + O(∆t2). i j 2 i j i j i j

Therefore,

∆t ν∆t ξ¯n+1/2 = ξn − {−δ ψn · δ ξn + δ ψn · δ ξn } + {δ2ξn + δ2ξn }· (2.16) ij ij 2 y ij x ij x ij y ij 2 x ij y ij

n+1/2 Then we construct ψij via step (b), i.e.,

2 2 n+1/2 2 2 n+1/2 2 2 n+1/2 2 ¯n+1/2 2 ¯n+1/2 δxδxψi,j +2δxδyψi,j + δyδyψi,j = δxξi,j + δyξi,j (2.17) for all interior points i =2, ··· ,M − 2,j=2, ··· ,K − 2. On the boundary and near-boundary points - the boundary conditions are imposed on ψ in the same way as in step (b) above. n+1/2 Finally, ξij is updated by step (c) as follows.

n+1/2 2 n+1/2 2 n+1/2 ξi,j = δxψi,j + δyψi,j (2.18) for all points i =0, ··· ,M, j =0, ··· ,K. n+1/2 n+1/2 Now, the approximated values of ξij and ψij are substituted in (2.15), which yields an implicit second order scheme for (2.2). From here, we proceed with steps (b) and (c) as before. The stability condition for this scheme, based on linear analysis, is max|u| max|v| ∆t + ≤ 1. ∆x ∆y VORTICITY DYNAMICS AND NUMERICAL RESOLUTION OF NAVIER-STOKES EQUATIONS 319

3. Numerical results We present here several problems, for which we applied our scheme. All the examples were studied using the explicit scheme. In the ﬁrst and second examples we measure the rate of convergence. The ﬁrst test problem has the following initial vorticity

ξ(x, y, 0) = π(cos(2πx)sin2(πy)+sin2(πx)cos(2πy)), (3.1) and streamfunction

ψ(x, y, 0) = (1/π)sin2(πx)sin2(πy), (3.2)

≤ ≤ ∂ψ for 0 x, y 1 [2]. The appropriate boundary conditions are ψ = ∂n =0onΓ. We have measured the numerical rate of convergence for ν =0.01, 0.001, 0.0001. For each ν we computed the solution on a uniform grid with h =∆x =∆y =1/2n,forn =4, 5, 6, 7tot =0.1. To estimate the rate of convergence, we compare the streamfunction, obtained on a grid with mesh size h, ψh, with the one obtained on the next ﬁner grid, ψh/2. k ave − k ave This is done by computing e(h)= ψh/2 ψh l2 . Here ψh/2 is the averaged value of ψh/2, computed at the coarser mesh points, and the average is taken on the coarse-point itself and its next four ﬁner-mesh neighbors. l2 indicates the discrete L2 norm. The results are represented in Table 1. Note that the rate of convergence is then given by e(h) p =log · 2 e(h/2)

Table 1.

mesh 16-32 Rate 32-64 Rate 64-128 ν =0.01 1.167 × 10−3 2.40 3.162 × 10−4 2.31 6.379 × 10−5 ν =0.001 2.031 × 10−3 2.06 4.842 × 10−4 2.12 1.115 × 10−4 ν =0.0001 2.066 × 10−3 2.02 5.063 × 10−4 2.04 1.229 × 10−4

The computed results indicate that the numerical rate of convergence is approximately 2. A second test problem has a known exact solution [6]

ξ(x, y, t)=e−2t sin x sin y, 0 ≤ x, y ≤ π. (3.3)

Table 2 displays the error e and the relative error er,where

e = kξcomp − ξexactkl2 , and er = e/kξexactkl2 . Here, ξcomp and ξexact are computed and exact vorticities, respectively. We present results for diﬀerent time-levels and number of mesh points, with ν =1. The third test problem, on which we have implemented the scheme, contains two concentrated vortices of opposite signs, located at two diﬀerent points of the domain Ω = [−5, 5] × [−5, 5]. Thus   {1 − 64[(x − 2)2 +(y − 2)2]}7, (x − 2)2 +(y − 2)2 ≤ 1/64, −{ − 2 2 }7 2 2 ≤ ξ(x, y, 0) =  1 64[(x +2) +(y +2) ] , (x +2) +(y +2) 1/64, (3.4) 0, otherwise.

Figures 1a–d represent the calculated vorticity at t =0.0078, 0.234, 0.391, 7.812, respectively, with a spatial mesh of 41 × 41 points. Similarly, Figures 2a–d show the time evolution of the streamfunction. Note that at the last step the vorticity and the streamfunction decay to the order of 10−6. 320 M. BEN-ARTZI ET AL. Table 2.

mesh M=K=10 Rate M=K=20 Rate M=K=40 t =1e 2.522 × 10−3 2.67 3.943 × 10−4 2.00 9.861 × 10−5 −2 −3 −4 er 1.151 × 10 1.849 × 10 4.630 × 10 t =2e 3.334 × 10−3 2.64 5.344 × 10−5 2.67 1.332 × 10−5 −2 −3 −4 er 1.152 × 10 1.849 × 10 4.630 × 10 t =3e 4.632 × 10−5 2.67 7.242 × 10−6 2.00 1.806 × 10−6 −2 −3 −4 er 1.152 × 10 1.849 × 10 4.630 × 10 t =4e 6.124 × 10−6 2.64 9.815 × 10−7 2.64 2.477 × 10−7 −2 −3 −4 er 1.152 × 10 1.849 × 10 4.630 × 10

"fort.40" "fort.40" 0.332 0.0131 0.166 0.00656 0 1.77e-06 -0.166 -0.00655 -0.332 -0.0131

0.5 0.02 0.4 0.015 0.3 0.01 0.2 0.005 0.1 0 0 -0.1 -0.005 -0.2 -0.01 -0.3 -0.015 -0.4 -0.5 -0.02 5 5

-5 0 -5 0 0 0 5 -5 5 -5 (a) t =0.0078 (b) t =0.234

"fort.40" "fort.40" 0.0074 1.14e-06 0.0037 5.8e-07 2.52e-06 1.5e-08 -0.0037 -5.5e-07 -0.0074 -1.11e-06 0.015 2e-06 0.01 1.5e-06 1e-06 0.005 5e-07 0 0 -5e-07 -0.005 -1e-06 -0.01 -1.5e-06 -2e-06 -0.015 5 5

-5 -5 0 0

0 0 5 -5 5 -5 (c) t =0.391 (d) t =7.812

Figure 1. Vorticity contours for two opposite-sign vortices.

Next, we show numerical results for a driven cavity with ν =1/400. Here the domain is Ω = [0, 1] × [0, 1] and the ﬂuid is driven in the x−direction on the top section of the boundary (y =1).Thus,u =1,v=0for y =1,andu = v =0forx =0,x=1andy = 0. In Table 3 we present computational quantities for diﬀerent meshes and time-levels. We show max |ψ|,(¯x, y¯), where (¯x, y¯)isthepointwheremax|ψ| occurs, and the value of the vorticity at these points. The meshes are of 129 × 129, 193 × 193 and 257 × 257 points and the time levels are t =9, 15, 30, 45. Note that the highest absolute value of the streamfunction at the latest time step is 0.1128. Here the maximum occurs at (¯x, y¯)=(0.5664, 0.5781), where the value of the vorticity is 2.3185. Note VORTICITY DYNAMICS AND NUMERICAL RESOLUTION OF NAVIER-STOKES EQUATIONS 321 Table 3.

time quantity 129 × 129 193 × 193 257 × 257 9 max |ψ| 0.1224 0.1177 0.1151 (¯x, y¯) (0.5703, 0.5859) (0.5729, 0.5885) (0.5742, 0.5859) ξ(¯x, y¯) 2.8493 2.7416 2.6879 15 max |ψ| 0.1223 0.1169 0.1139 (¯x, y¯) (0.5625, 0.5859) (0.5677, 0.5803) (0.5664, 0.5781) ξ(¯x, y¯) 2.6083 2.4630 2.3964 30 max |ψ| 0.1217 0.1159 0.1128 (¯x, y¯) (0.5625, 0.5859) (0.5625, 0.5781) (0.5664, 0.5781) ξ(¯x, y¯) 2.5520 2.4016 2.3196 45 max |ψ| 0.1214 0.1159 0.1128 (¯x, y¯) (0.5625, 0.5859) (0.5625, 0.5781) (0.5664, 0.5781) ξ(¯x, y¯) 2.5460 2.4003 2.3185

"fort.41" "fort.41" 0.0174 0.00732 0.00868 0.00366 -6.93e-06 -4.26e-06 -0.00869 -0.00367 -0.0174 -0.00733 0.015 0.03 0.01 0.02 0.005 0.01 0 0

-0.01 -0.005 -0.02 -0.01 -0.03 -0.015 5 5

-5 0 -5 0

0 0 -5 5 -5 5 (a) t =0.0078 (b) t =0.234

"fort.41" "fort.41" 0.00553 1.39e-06 0.00276 6.73e-07 -6.08e-06 -4e-08 -0.00278 -7.53e-07 -0.00555 -1.47e-06 0.01 2.5e-06 2e-06 1.5e-06 0.005 1e-06 5e-07 0 0 -5e-07 -1e-06 -0.005 -1.5e-06 -2e-06 -2.5e-06 -0.01 5 5

-5 -5 0 0

0 0 5 -5 5 -5 (c) t =0.391 (d) t =7.812

Figure 2. Streamfunction contours for two opposite-sign vortices. also that the location (¯x, y¯) for the finest grid has been stabilized at t = 15. In [15] max |ψ| =0.1139 occurs at (0.5547, 0.6055), where the value of the vorticity is 2.2947. In Table 4 we display the same flow quantities as in Table 3, but for ν =1/3200 at t =60, 120, 150, 180. The meshes are of 193 × 193, 257 × 257 and 513 × 513 points. Note that with each of the meshes the flow quantities tend to converge to a steady state as time progresses. At the latest time level on the finest grid the maximal value of |ψ| is 0.1246, which is obtained at (¯x, y¯)=(0.5254, 0.5078), and ξ(¯x, y¯)=2.0447. The values reported in [15] are max |ψ| =0.1204, obtained at (0.5165, 0.5469), with ξ =1.9886. Figures 3a–d and 4a–d display the 322 M. BEN-ARTZI ET AL.

0.01294 500 0.01363 0.01363 500 0.01294 0.01294 0.01363 0.02726 0.02726 0.02588 0.04089 0.03882 0.03882 0.12 0.02588 0.04089 0.05452 0.05176 450 0.12 450 0.05176 0.06815 0.0647 0.02726 0.03882 0.08178 0.05452 0.07764 400 0.05452 0.09541 400 0.0647 0.07764 0.1 0.05176 0.1 0.09058 0.09058 0.06815 0.10904 0.10352 350 0.08178 350 0.11646 0.12267 0.01363

0.01294 0.01294 0.08 0.01363 300 0.08 300

0.04089 0.12267 0.10904

0.07764 0.02588 0.09541

0.09058

0.05452

0.0647 250 0.02726 250 0.11646 0.05176

0.10352

0.02726 0.06815 0.03882 0.06 0.06 0.08178

0.05176 200 200 0.02588

0.01363 0.01294 0.10904 0.09541 0.04 150 0.04089 0.09541 0.04 150 0.10352 0.02588 0.09058 0.01363 0.07764 0.07764 0.08178 0.06815 0.03882 0.01294 100 0.02726 0.05452 0.06815 100 0.0647 0.03882 0.05176 0.0647 0.02 0.04089 0.02 −2.16e−05 0.05452 0.02726 0.05176 0.04089 0.03882 50 0.01363 0.02726 50 0.01294 0.02588 0.02588 0.01363 0.01294 0 0 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 (a) t =60 (b) t = 120

0.01264 0.01264 0.01252 0.01252 500 0.02528 500 0.01252 0.02528 0.12 0.02504 0.02504 0.03792 0.03792 0.03756 0.03756 0.02528 0.05056 0.05008 0.1 450 0.05056 450 0.05008 0.01264 0.0632 0.03756 0.0626 0.0632 0.0626 0.07512 0.03792 0.07584 400 0.08848 0.1 400 0.02504 0.07584 0.08764 0.08764 0.10112 0.08 0.07512 0.10016 350 350 0.10016 0.0632 0.11376 0.05008 0.08 0.01252

0.11268 0.01252 0.08848 300 0.05056 300 0.02504

0.0626

0.03792 0.02528 0.08764 0.06

0.07584 0.11376 0.02504 250 0.06 250 0.03756

0.10112 0.01264 0.10016

200 0.02528 200 0.07512 0.01264 0.0632 0.11268 0.05008 0.04 0.01252 0.0626 0.08848 0.04 0.07512 150 0.05056 0.10112 150 0.10016 0.0626 0.02528 0.08848 0.07584 0.0632 0.08764 0.07584 0.03792 0.03756 0.07512 0.01252 100 0.01264 0.03792 0.05056 100 0.03756 0.02504 0.02 0.0632 0.02 0.0626 0.05008 0.05056 0.02504 0.05008 0.03792 0.03756 50 0 0.02528 0.02528 50 0.01252 0.01264 0.02504 0.01264 0.01252 0 0 0 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 (c) t = 150 (d) t = 180

Figure 3. Driven cavity for ν =1/3200: streamfunction contours.

Table 4.

time quantity 193 × 193 257 × 257 513 × 513 60 max |ψ| 0.2007 0.1766 0.1364 (¯x, y¯) (0.5260, 0.5156) (0.5273, 0.5156) (0.5332, 0.5098) ξ(¯x, y¯) 4.5183 3.8447 2.7479 120 max |ψ| 0.2017 0.1734 0.1264 (¯x, y¯) (0.5260, 0.5156) (0.5273, 0.5156) (0.5254, 0.5078) ξ(¯x, y¯) 4.4479 3.5662 2.1420 150 max |ψ| 0.2018 0.1730 0.1252 (¯x, y¯) (0.5260, 0.5156) (0.5273, 0.5156) (0.5254, 0.5078) ξ(¯x, y¯) 4.4535 3.5434 2.0736 180 max |ψ| 0.2019 0.1726 0.1246 (¯x, y¯) (0.5260, 0.5156) (0.5273, 0.5156) (0.5254, 0.5078) ξ(¯x, y¯) 4.4563 3.5297 2.0447

streamfunction and the vorticity contours, respectively, at t =60, 120, 150, 180, using a 513 × 513 mesh. Note that the transients to steady state are smooth. In Figures 5a–b we present velocity components u(0.5,y)and v(x, 0.5) (solid lines) compared with values obtained in [15] (marked by “+”), for ν =1/3200 at t =90, 180, respectively. VORTICITY DYNAMICS AND NUMERICAL RESOLUTION OF NAVIER-STOKES EQUATIONS 323

500 200 500 200

450 450 100 100 400 400

350 0 350 0

300 300 −100 −100 250 250

200 −200 200 −200

150 150 −300 −300 100 100

50 −400 50 −400

50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 (a) t =60 (b) t = 120

500 200 500 200

450 450 100 100 400 400

350 0 350 0

300 300 −100 −100 250 250

200 −200 200 −200

150 150 −300 −300 100 100

50 −400 50 −400

50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 (c) t = 150 (d) t = 180

Figure 4. Driven cavity for ν =1/3200: vorticity contours.

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

−0.2 −0.2

−0.4 −0.4

−0.6 −0.6

−0.8 −0.8

−1 −1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (a) ν =1/3200,t=90 (b) ν =1/3200,t= 180

Figure 5. Velocity components in the driven cavity (solid lines), compared with Ghia et al. (marked “+”).

A similar problem, that we considered, has the same geometry Ω, but here the ﬂuid is driven also in the negative y−direction at the right-end of Ω. Thus, u = −1,v=0fory =1,u =0,v= −1forx =1, u = v =0forx =0andy = 0. We picked ν =1/1000 and a 81 × 81 mesh. Figures 6a–d and 7a–d represent the streamfunction and the vorticity evolution, respectively, at t =0.04, 0.39, 3.90 and 7.81. 324 M. BEN-ARTZI ET AL.

8 −0.0037895 −0.0037895 −0.034106 −0.003836 −0.003836 −0.003836 80 −0.011368 −0.011368 80 −0.011508 −0.011508 −0.026852 −0.011508 −0.018947 −0.026526 −0.018947 −0.01918 −0.01918 −0.026526 −0.034106 −0.026852−0.034524 −0.026852−0.034524 −0.049263 −0.041684 −0.049868−0.042196 −0.056843 −0.064421 0.06 −0.05754 0.06 −0.072001 −0.0037 −0.065212 −0.072884 −0.011368 70 70 −0.01918 0.011508 0.0037895 −0.011368

−0.018947 0.04 −0.003836 0.04 −0.042196−0.034524 0.026852 −0.049263 −0.026526 60 −0.041684 0.026526 60 −0.049868 −0.011508 −0.034106 0.018947 −0.056843 −0.003836 −0.065212 −0.05754 0.003836 0.011508 −0.018947 −0.049263 −0.049868 0.011368 −0.0037895 −0.041684 0.041684 0.02 0.01918 0.02 −0.034106 −0.026852 −0.042196 50 50 −0.026852 −0.01918 0.034106 0.018947 −0.011368 −0.034524

−0.0037895 0.0037895 0.034106 −0.011508 0.042196 −0.026526 0.056843 −0.01918 0.026852 0.05754 0

0.011368

−0.018947 003836 −0.003836 0.034524 0 −0.011508 0.049868 0 40 0.064421 40 −0.011368 0.065212

0.041684 0.003836 0.011508 0.026526 0.018947 −0.0037895 0.026852 0.011368 0.034106 −0.003836

30 0 0037895 −0.02 30 −0.02 0.049263 0.01918 0.011508 0.072001 0.05754 0.042196 0.056843 0.026852 0.034524 0.049868 20 0.0037895 20 0.064421 −0.04 −0.04 0.003836 0.065212 0.026526 0.011508 0.018947 0.049263 0.011368 0.026526 0.05754 0.041684 0.01918 0.049868 10 0.034106 10 0.0037895 68 −0.06 −0.06 0.018947 0.026852 0.042196 03836 89 0.003836 0.011508 0.034524 0.01918

0.0113 0.0 0.0037 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 (a) t =0.04 (b) t =0.39

−0.0053083 −0.0053083 0.1 −0.005674 −0.005674 80 −0.015925 −0.015925 80 −0.017022 −0.017022 −0.026542 −0.026542 −0.02837 0.1 −0.037158 −0.039718 −0.047775 −0.051066 −0.058392 −0.0053083 0.08 −0.062415 −0.069008 −0.037158 70 70 −0.062415 −0.085111 −0.073763 0.08 −0.0053083 −0.005674 0 0.005674

−0.079625 0053083 −0.096459 −0.090242 −0.10781 0.017022 −0.015925 0.06 0

−0.015925 . 0.06

005674 60 60 −0.02837 −0.026542 0.0053083 0.04 0.04 −0.051066 −0.058392 −0.017022 0.039718 −0.037158 −0.0053083 0.026542 −0.005674 50 0.015925 50 −0.017022 −0.039718 0.02837 −0.090242 −0.047775 −0.026542 0.037158 0.02 0.02 −0.02837 −0.096459 0.005674 −0.005674 0.062415 0.017022 0.017022 −0.069008 −0.058392 0.047775 −0.085111 0.058392 0.073763 0.051066 −0.073763 −0.047775 −0.015925 0 0 40 40 −0.062415 0.051066 −0.037158 0.037158 0.069008 0.0053083 −0.039718−0.051066 −0.02837

−0.0053083 0.026542 0.047775 0

−0.026542 0.090242 . 0.039718 −0.015925 0053083 −0.02 −0.02 −0.017022

−0.0053083 0.10086 0.062415

30 30 0.096459 005674 0.015925 0.10781 0.015925 0.058392 0.079625 0.026542 −0.005674 0.02837 0.073763 −0.04 −0.04 0.005674 0.085111 0.0053083 0.037158 0.069008 0.051066 20 20 0.017022 0.090242 −0.06 0.026542 0.047775 −0.06 0.079625 0.039718 0.085111 0.062415 0.073763 0.058392 −0.08 10 0.015925 −0.08 10 0.037158 0.047775 0.02837 0.051066 0.026542 0.005674 0.017022 0.039718 0.0053083 0.015925 0.02837 −0.1 3 0.005674 0.017022 0.005308 −0.1 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 (c) t =3.90 (d) t =7.81

Figure 6. Double-driven cavity for ν =1/1000: streamfunction contours.

80 80 12.4337 19.2157214.69438 100 7.912356 10.17303 100 5.651683 3.321446 7.75004 5.535743 5.651683 5.535743 3.39101 5.535743 70 1.107149 80 70 −21.4764 80 3.321446 5.651683

3.39101 7.912356 1.130337

10.17303 3.321446 1.107149 60 3.39101 60 60 60 1.130337

40 40 −1.130337 1.130337 −1.107149 50 50 −10.17303 1.107149 20 20 −7.912356

−5.535743 −3.321446 −1.130337 −3.39101 40 0 40 0

−1.130337 −20 −20

−1.107149

30 30 −5.651683 −40 −40

−3.39101 −3.321446 20 −60 20 −60

−5.651683 −7.75004 1.130337 −80 −5.535743 −80 10 10 −1.130337 3.39101 5.651683 −100 −100 10.17303−7.912356

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 (a) t =0.04 (b) t =0.39

80 23.73624 25.99683 80 21.47546 23.73603 10.17267 7.912079 12.43327 16.95446 14.6938612.43327 19.21505 10.17258 7.91201 10.17258 19.21488 14.69373 5.651485 5.651485 10.17267 7.912079 5.651485 100 5.651436 5.651436 7.91201 100 3.390891 3.390891 3.390861 3.390861 3.390891 1.130297 1.130297 1.130297

−16.95431

1.130297 −10.17258 70 3.390891 7.912079 80 70 3.390861 80 3.390891

−19.21488 −3.390861 −16.95446

−1.130297

60 −1.130287 60

−12.43316

−14.69386 −7.912079

60 −12.43327 60 1.130287 −10.17258 −7.91201 3.390891 40 40 −12.43316

3.390891 −3.390891 1.130287 −1.130287 50 5.651485 50 20 20 −1.130297 −1.130297 −3.390861 3.390861 −10.17267 −5.651485 3.390861 1.130297 3.390891 − 1.130297 0 −5.651436 0 40 3.390891 40 −3.390861 −3.390891 −7.912079 −12.43327 1.130287 −3.390861 −7.91201 −20 −1.130287 −20 −1.130297 30 30 −1.130297 −1.130287 −40 −40 1.130287 1.130297 −3.390891

−1.130297 −3.390861 20 −5.651485 −60 20 −60 −3.390861 −3.390891 −3.390891

−10.17258 −1.130297 −80 −80 10 −3.390891 10 −1.130287 −5.651485 1.130297 −3.390891 1.130287 −7.91201 5.651485 3.390891−1.130297−10.17267 −100 3.390861−5.651436−3.390861 −100 7.912079 10.17267 16.95446 16.954 −7.912079 31 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 (c) t =3.90 (d) t =7.81

Figure 7. Double-driven cavity for ν =1/1000: vorticity contours. VORTICITY DYNAMICS AND NUMERICAL RESOLUTION OF NAVIER-STOKES EQUATIONS 325

250 250

300 300

200 200 200 200

100 100 150 150

0 0

100 100 −100 −100

−200 −200 50 50

−300 −300

50 100 150 200 250 50 100 150 200 250 (a) t =1.5 (b) t =3

250 250

300 300

200 200 200 200

100 100 150 150

0 0

100 100 −100 −100

−200 −200 50 50

−300 −300

50 100 150 200 250 50 100 150 200 250 (c) t =18 (d) t =45

Figure 8. Double-driven cavity for ν =1/10000: vorticity contours.

In Figures 8a–d we display vorticity contours at t =1.5, 3, 18, 45 respectively for ν =1/10000 with a 257×257 grid. Note that the computational results remain stable, retain the symmetry along y = x and capture small scale phenomenon at high Reynolds numbers. In [27] a Hopf bifurcation and break of symmetry are observed at ν =1/5000, but we could not confirm it. The differences may be explained by the observation that in [27] the initial condition was taken as the steady-state solution of the same problem with a lower Reynolds number. Here the flow starts impulsively from zero and a steady-state is not yet fully reached. We further demonstrate results for a flow over a square cylinder. The computational domain is 0 ≤ x ≤ 12, 0 ≤ y ≤ 3, and the square cylinder lies at 8 ≤ x ≤ 9, 1 ≤ y ≤ 2. On the outer boundaries we impose u = −1,v= 0, and on the inner boundary (the cylinder) u = v = 0. We picked ν =10−3. Figures 9a–d display the velocity field at t =2.5, 5, 7.5, 10 respectively, using a 121 × 31 mesh. A Karman vortex street develops as time evolves (see [21] for computations using a control volume formulation, [20] for computations including ground effects and [25] for experimental results). Our last example is a backward facing step, so that the inflow section is smaller than the outflow one. We thank O. Pironneau for suggesting to us this example, including the initial data. This geometry produces a fluid recirculation zone, that is clearly captured in the results. The setup is shown in Figure 10. We impose a parabolic inflow at the entrance (x =0),i.e., u =1− y2,v= 0 and a parabolic outflow at the exit (x = 12), i.e., u = C(1 − y)(y + 2). The constant C is determined such that the flux at the entrance equals the flux at the exit. Thus C =8/27. We picked ν =1/1000. In Figures 11a–d we show contours of the horizontal velocity u at times 1.25, 2.5, 5, 7.5 on a 121 × 31 mesh. Then, for the same times, we show the corresponding contours of the streamfunction in Figures 12a–d, and the corresponding velocity maps in Figures 13a–d. Similar computations were done for the physical domain 0 ≤ x ≤ 30, −2 ≤ y ≤ 1, where the step lies in 0 ≤ x ≤ 3, −2 ≤ y ≤−1 with ν =1/150. Figures 14a–d 326 M. BEN-ARTZI ET AL.

51 51

50.5 50.5

50 50

49.5 49.5

49 49

48.5 48.5

48 48 50 52 54 56 58 60 62 50 52 54 56 58 60 62 (a) t =2.5 (b) t =5

51 51

50.5 50.5

50 50

49.5 49.5

49 49

48.5 48.5

48 48 50 52 54 56 58 60 62 50 52 54 56 58 60 62 (c) t =7.5 (d) t =10

Figure 9. Velocity ﬁeld for a ﬂow over a square cylinder.

y 12 L

x 2 L

2 L L

10 L

Figure 10. A backward facing step. VORTICITY DYNAMICS AND NUMERICAL RESOLUTION OF NAVIER-STOKES EQUATIONS 327

30 0.047156 30 0.065315 0.065315 0.11396 0.11396 0.27432 0.13498 0.13498 0.24757 0.18077 0.18077 0.20465 0.20465 0.24757 0.24757 0.34398 0.5148 0.27432 0.27432 0.38119 0.31438 0.31438 0.55299 0.71522 0.76199 0.34398 0.38119 0.41365 25 0.64841 25 0.48332 0.41365 0.91564 0.44799 0.44799 0.78203 0.83166 0.48332 0.98244 0.5148 0.5148 0.97099 0.55299 0.55299 0.58161 0.58161 0.62265 0.69232 0.62265 20 20 0.64841 0.64841 0.97099 0.91564

0.78203 0.84883 0.83166 0.90132 0.71522 0.76199 15 15 0.69232 0.48332 0.62265 0.24757 0.5148 0.64841 0.64841 0.62265 0.047156 0.31438 0.62265 0.58161 0.20465 0.34398 10 0.58161 10 −0.0043525 0.55299 0.44799 0.55299 0.11396 0.5148 0.5148 0.38119 0.065315 0.41365 0.27432 0.48332 0.48332 0.44799 0.41365 0.18077 0.38119 0.13498 0.34398 0.34398 5 0.31438 0.31438 5 0.24757 0.24757 0.27432 0.18077 0.18077 0.20465 0.20465 0.11396 0.11396 0.13498 0.047156 0.047156 0.065315 0.065315

20 40 60 80 100 120 20 40 60 80 100 120 (a) t =1.25 (b) t =2.5

30 30 0.11813 0.2452 0.084655 0.19567 0.11813 0.084655 0.16493 0.19567 0.32548 0.50584 0.16493 0.2452 0.35076 0.27322 0.5663 0.27322 0.32548 0.73847 0.35076 0.40575 0.66093 0.80712 0.48602 0.2452 25 0.81601 0.4283 25 0.32548 0.40575 0.4283 0.88739 0.9711 0.50584 0.64657 0.58339 0.72685 0.48602 0.50584 0.5663 0.58339 0.5663 0.89356 20 0.73847 20 0.96767 0.80712 0.64657 0.66093 0.66093 0.96767 0.89356 0.9711

0.80712 0.72685 0.73847 0.88739 15 15 0.64657 0.81601 0.48602 0.50584 0.66093 0.19567 0.66093 0.72685 0.64657 0.040588 0.27322 0.58339 0.0043813 0.16493 0.32548 0.5663 0.64657 0.4283 0.58339 10 0.35076 10 0.40575 0.5663 0.2452 0.48602 0.11813 0.50584 0.48602 0.50584 0.084655 0.4283

0.19567 0.0043813 0.40575 0.35076 0.32548 5 0.27322 5 0.32548 0.27322 0.040588 0.2452 0.19567 0.16493 0.11813 0.11813 0.16493 −0.036955 −0.075893 0.084655 0.040588 0 0043813 20 40 60 80 100 120 20 40 60 80 100 120 (c) t =5 (d) t =7.5

Figure 11. Horizontal velocity in backward facing step for ν =1/1000.

30 30

−1.2495 −1.2475 −1.1657 −1.2495 −1.2495 −1.1617 −1.2475 −1.2475 25 25 −1.0759 −0.99799 −1.1657 −1.1657 −1.1617 −1.1617 −0.91416 −1.0818 −0.83033 −0.90434 −1.0818 −0.81854 −0.99013 −1.0759 −1.0759 −0.73274 −0.99799 −0.99799 −0.99013 20 −0.91416 −0.91416 20 −0.66266 −0.90434 −0.90434 −0.83033 −0.56114 −0.74649 −0.83033 −0.81854 −0.81854 −0.49499 −0.74649 −0.64694 −0.41116 −0.38954 −0.73274 −0.73274 −0.66266 −0.66266 −0.30374 −0.64694 15 −0.15966 −0.57883 15 −0.57883 −0.56114 −0.56114 −0.24349 −0.49499 −0.49499 −0.046349 −0.47534 −0.13215 −0.47534 −0.32732 −0.41116 −0.41116 −0.38954 −0.38954 −0.21795 −0.32732 −0.30374 10 10 −0.30374 −0.075824 −0.24349 −0.24349 −0.21795 −0.15966 −0.15966 −0.13215 −0.13215 −0.075824 −0.075824 5 5 −0.046349 −0.046349

20 40 60 80 100 120 20 40 60 80 100 120 (a) t =1.25 (b) t =2.5

30 30

−1.2442 −1.2437 −1.1551 −1.1541 −1.2442 −1.2442 −1.2437 25 −1.066 25 −1.0644 −1.2437 −1.1551 −0.97478 −1.1541 −0.88782 −1.1551 −0.97692 −1.1541 −0.79871 −1.066 −0.79551 −0.88514 −1.0644 −0.70961 −1.066 −0.70587 −0.97692 −0.61623 −1.0644 −0.97478 20 20 −0.97478 −0.53141 −0.88782 −0.88782 −0.43695 −0.88514 −0.79871 −0.79551 −0.62051 −0.79871 −0.52659 −0.3532 −0.70961 −0.70961 −0.79551 −0.70587 −0.2641 −0.25768 −0.61623 −0.70587 −0.16804 15 −0.62051 15 −0.61623 −0.4423 −0.34732 −0.085895 −0.53141 −0.53141 −0.52659 −0.4423 −0.43695 −0.175 −0.078405 −0.43695 −0.3532 −0.3532 −0.34732 0.0032075 10 −0.2641 10 −0.25768 −0.2641 −0.25768 −0.175 −0.16804 0.0032075 −0.16804

−0.085895 −0.085895 −0.078405 5 5

0.011233

20 40 60 80 100 120 20 40 60 80 100 120 (c) t =5 (d) t =7.5

Figure 12. Streamfunction in backward facing step for ν =1/1000. 328 M. BEN-ARTZI ET AL.

51 51

50.5 50.5

50 50

49.5 49.5

49 49

48.5 48.5

48 48 50 52 54 56 58 60 62 50 52 54 56 58 60 62 (a) t =1.25 (b) t =2.5

51 51

50.5 50.5

50 50

49.5 49.5

49 49

48.5 48.5

48 48 50 52 54 56 58 60 62 50 52 54 56 58 60 62 (c) t =5 (d) t =7.5

Figure 13. Velocity ﬁeld in backward facing step for ν =1/1000.

1 1

−1.2726 −1.2726 −1.2726 −1.2726 0.5 −1.2726 0.5 −1.212 −1.212 −1.2118 −1.2118 −1.2118 −1.0299 −1.1513 −1.1513 −1.1513 −1.151 −0.8472 −1.151 −1.0906 −1.0906 −1.0906 −1.0903 −1.0903 −1.0903 0 −1.0299 −1.0299 0 −1.0295 −1.0295 −0.96921 −0.96921 −0.96873 −0.96873 −0.90852 −0.90852 −0.90852 −0.90796 −0.90796 −0.90796 −0.60508 −0.72646 −0.84783 −0.84783 −0.84783 −0.8472 −0.8472 −0.42183 −0.78715 −0.78715 −0.78715 −0.72566 −0.78643 −0.78643 −0.66577 −0.72646 −0.72646 −0.72566 −0.72566 −0.5 −0.30165 −0.66577 −0.66577 −0.5 −0.6649 −0.6649 −0.6649 −0.60508 −0.60508 −0.36106 −0.60413 −0.057226 −0.60413 −0.5444 −0.5444 −0.5444 −0.54336 −0.54336 −0.24096 −0.42302 −0.48371 −0.4826 −0.4826 −0.48371 −0.17876 −0.42302 −0.42302 −0.4826 −0.42183 −0.42183 −0.36234 −0.36234 −0.36234 −0.36106 −0.36106 −1 −1 −0.30165 −0.30165 −0.30029 −0.30029 −0.058898 −0.24096 −0.24096 −0.23953 −0.23953 −0.11959 −0.18027 −0.18027 −0.18027 −0.17876 −0.17876 −0.11959 −0.11959 −0.11799 −0.11799 −0.11799 −1.5 −1.5 −0.058898 −0.058898 −0.057226 −0.057226 0.0017889

0.0035406 −2 −2 0 5 10 15 20 25 30 0 5 10 15 20 25 30 (a) t =5 (b) t =7.5

1 1

−1.2442 −1.2265 −1.2888 −1.2888 −1.2799 −1.2799 −1.1996 0.5 −1.173 0.5 −1.2442 −1.2442 −1.2265 −1.2265 −1.155 −1.1196 −1.1105 −1.1996 −1.1996 −1.0661 −1.173 −1.173 −0.93219 −0.97676 −1.155 −1.155 −0.85239 −1.1196 −1.1196 −1.0213 −0.90583 −1.1105 −1.1105 −0.95926 −1.0661 −1.0661 −1.0659 −1.0659 0 −1.0127 −1.0127 0 −1.0213 −1.0213 −0.95926 −0.95926 −1.0659 −0.97676 −0.97676 −0.44189 −0.57561 −0.84304 −0.93219 −0.93219 −1.0127 −0.90583 −0.90583 −0.88761 −0.74551 −0.85239 −0.85239 −0.70933 −0.7539 −0.84304 −0.84304 −0.26457 −0.42488 −0.79895 −0.79847 −0.5852−0.63863 −0.74551 −0.74551 −0.88761 −0.79847 −0.7539 −0.7539 −0.79895 −0.62018 −0.70933 −0.70933 −0.69207 −0.69207 −0.5 −0.63863 −0.63863 −0.5 −0.35275 −0.66475 −0.47832 −0.62018 −0.62018 −0.5852 −0.5852 −0.085319 −0.66475 −0.57561 −0.57561 −0.53176 −0.12989 −0.53104 −0.53104 −0.53104 −0.040747 −0.26361 −0.15769 −0.53176 −0.47832 −0.47832 −0.17446 −0.48647 −0.42488 −0.42488 −0.39732 −0.44189 −0.37144 −0.48647 −0.44189 −0.39732 −0.39732 −0.37144 −0.37144 −0.35275 −0.35275 −1 −1 −0.318 −0.21903 −0.30818 −0.050812 −0.21113 −0.318 −0.30818 −0.26457 −0.26457 −0.26361 −0.26361 −0.21113 −0.21113 −0.21903 −0.21903 −0.15769 −0.15769 −0.17446 −0.17446 −0.12989 −0.12989 −0.10425 −0.10425 −0.10425 −0.085319 −1.5 0.0026262 −1.5 −0.085319 0.0038249 −0.050812 −0.050812 −0.040747 −0.040747

−2 −2 0 5 10 15 20 25 30 0 5 10 15 20 25 30 (c) t =10 (d) t =12.5

Figure 14. Streamfunction contours in backward facing step for ν =1/150. VORTICITY DYNAMICS AND NUMERICAL RESOLUTION OF NAVIER-STOKES EQUATIONS 329 represent streamfunction contours at t =5, 7.5, 10, 12.5 respectively. Note that the length of the wake behind the cylinder is approximately of 8.2 length units. The latter compares favorably with computations done by Glowinski [16].

Acknowledgements. It is a pleasure to thank Professors A. Chorin, R. Glowinski, S. Kamin, O. Pironneau and R. Temam for very useful discussions concerning this paper as well as the broader issues of numerical resolution of the Navier- Stokes equations. We are grateful to Professor Alexandre Ern for pointing out to us the works of Quartapelle and his collaborators, and to Professors Christophe Besse and Komla Domelevo for pointing out an error in an earlier version of the paper.

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