FEM solution of the ψ-ω equations with explicit viscous diffusion 1

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1 FEM solution of te ψ-ω equations wit explicit viscous diffusion J.-L. Guermond and L. Quartapelle 3 Abstract. Tis paper describes a variational formulation for solving te D time-dependent incompressible Navier Stokes equations expressed in terms of te stream function and vorticity. Te difference between te proposed approac and te standard one is tat te vorticity equation is interpreted as a dynamical equation governing te evolution of (te weak Laplacian of) te stream function wile te Poisson equation for te stream function is used as an expression to evaluate te distribution of te vorticity in te domain and on te boundary. A time discretization is adopted wit te viscous diffusion made explicit, wic leads to split te viscous effects from te incompressibility similarly to te fractional-step projection metods for te primitive variable equations. In some sense, te present metod generalizes to te variational framework a well-known idea tat is used in nite differences approximations and wic is based on a Taylor series expansion of te stream function on te boundary. Some error estimates and some numerical results are given. INTRODUCTION Te prototype nite element procedure for solving te D Stokes equations formulated in terms of te vorticity and stream function is te uncoupled solution metod for te biarmonic problem introduced by Glowinski and Pironneau []. Tis approac can compute te solution of te Stokes problem by an uncoupled direct metod. in Proc. Computational Fluid Dynamics 98, ECCOMAS96, (Septembre 998, Atne, Grce), K.D. Papailiou, D. Tsaalis, J. Priaux, M. Pandol (Eds.), Wiley, Vol. (998) Laboratoire d Informatique pour la Mécanique et les Sciences de l Ingénieur, CNRS, BP 33, 93, Orsay, France (guermond@limsi.fr). 3 Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci, 3, 33 Milano, Italy. In te case of te time-dependent equations, tis metod assumes necessarily an implicit treatment of te viscous term and determines te unknown boundary value of te vorticity by means of an operator associated wit tat part of te boundary were no-slip velocity conditions are prescribed. Wile te tecniques based on tis metod or on te related idea of an in uence matrix guarantee good stability properties, tey are not so easy to implement; ence one may be tempted to make a trade-off between stability and simplicity. In te present paper we investigate one possible alternative tecnique for solving te evolutionary Navier Stokes equations in two dimensions expressed in a weak variational form. In te context of nite differences, wic are known for teir simplicity, one classical approac is to assume an explicit treatment of te viscous term to derive vorticity boundary formulas. In tis case, te Neumann condition for te stream function can be used as te last piece of te time-stepping algoritm. More precisely, te derivative boundary condition can be interpreted as a relationsip specifying te boundary value of te new vorticity after te time advancement of te (internal distribution of) vorticity as been completed and after te new stream function as been determined, for details see e.g. Peyret and Taylor [7], E and Liu [] or Napolitano et al. [6]. Te aim of tis paper is to develop te variational counterpart of tis idea and to sow tat an interpretation alternative to te standard one can be given to te vorticity equation written in weak form wic leads to a new and very simple algoritm for te numerical solution of te unsteady ψ-ω equations by means of nite elements. VARIATIONAL PROBLEM Te proposed numerical metod stems from te following variational statement of te incompressible Navier Stokes equation for D ows in a simply connected, c 998 J.-L. Guermond and L. Quartapelle ECCOMAS 98. Publised in 998 by Jon Wiley & Sons, Ltd.

2 bounded domain Ω wit a smoot boundary Γ = Ω. For te initial solenoidal velocity u belonging to H (Ω) and te body force f L (, T ; L (Ω)), we ave te problem Find ψ L (, T ; H (Ω)) C(, T ; H (Ω)) and ω L (, T ; L (Ω)) C(, T ; H (Ω)), so tat φ H (Ω), (( ψ) t=, φ) = (u, φ ẑ), and tat, for all t >, ( ψ H ψ (Ω), t, ψ ) + ν ω, ψ + b(ω, ψ, ψ ) = (f, ψ ẑ), v L (Ω), (ω, v) + ( ψ, v) =. () Homogeneous velocity boundary conditions for te sake of simplicity. Te trilinear form b(ω, ψ, ψ ) associated wit te advection term is written in rotational form, namely, b(ω, ψ, ψ ) = (ω ψ, ψ ẑ), () to guarantee conservation of te (kinetic) energy, wen ν =, also in te spatially discrete case of H - conformal approximations. Note te unusual form of te evolutionary term wic involves mixed time and space derivatives of ψ. Actually, tis weak form of te dynamical equation is te most natural one witin te present variational setting since it stems from te momentum equation were 3 SPATIAL DISCRETIZATION Let W and Ψ, be two nite dimensional subspaces of H (Ω) and H (Ω), respectively. We assume tat Ψ, W, and W, Ψ, satisfy standard interpolation and inverse properties (described in detail in [5]). FIRST ORDER SCHEME In tis section we approximate te time derivative by means of te rst order Euler sceme. Let [, T ] be a nite time interval and N be an integer. We denote δt = T/N and t n = nδt for n N. For any function of time, ϕ(t), we denote ϕ n = ϕ(t n). Te fully discrete problem is formulated as follows. Te initialization step reads: Find ψ Ψ, so tat, φ Ψ,, (3) ( ψ, φ ) = (u, φ ẑ), Find ω W so tat, v W, (ω, v ) = ( ψ, v ). Ten for eac n, carry out te following two uncoupled steps: Find ψ n+ Ψ, so tat, φ Ψ,, ( (ψ n+ ψ), n φ ) δt + b(ω n, ψ n+, φ ) = ν ( ω n, φ ) + (f n+, φ ẑ), () (5) te velocity is replaced by ψ ẑ, te viscous term is written as te curl of ω, and te velocity test functions belonging to V = v H (Ω), v = } and Find ω n+ W so tat, v W, (ω n+, v ) = ( ψ n+, v ). (6) are expressed as ψ ẑ by virtue of te te wellknown isomorpim (see Girault and Raviart [3])... ẑ : H (Ω) V. At variance wit more usual ways of writing te vorticity transport equation, no integration by parts is to be performed; in oter words, te curl of te momentum equation as not been taken in a strong form. In tis way te momentum equation becomes an evolutionary equation for te stream function (actually for its Laplacian in weak form) wereas te Poisson equation is used as te de nition of vorticity. Te nonlinear term is accounted for in a semi-implicit form for te sake of simplicity. All tat is said afterwards olds wit minor modi cations if tis term is made explicit. From te teoretical point of view, te modi cations in question essentially amount to deriving sligtly sarper bounds for te nonlinear residuals. Observe tat, wen compared to te classical ω-ψ formulation, te present metod intercanges te role of te variables ψ and ω. Here, te dynamical equation for te transport of ω as turned into an equation governing te evolution of (te weak form of) ψ, wereas te Poisson equation for ψ as become an expression controlling ω explicitly. Explicit ψ-ω metod 59 J.-L. Guermond and L. Quartapelle

3 . Vorticity integral conditions Note tat te explicit evaluation of te new vorticity eld ω n+ troug te solution of te mass matrix problem (6) does enforce te integral conditions for te vorticity wic underlay te uncoupled metod due to Glowinski and Pironneau []. In fact, considering more general, i.e., nonomogeneous, boundary conditions ψ n+ Γ = a n+ and ψn+ = b n+, n Γ te vorticity problem would read Find ω n+ W so tat, v W, (ω n+, v ) = ( ψ n+, v ) b n+ v. Selecting te functions v in te subspace of te discrete armonic functions, namely, v = η W so tat ( η, v ) =, v W, te weak equation above gives ( (ω n+, η ) a n+ η n ) bn+ η, Γ since it can be sown tat Γ an+ η / n ( ψ n+, η ). One recovers te vorticity integral conditions for te transient problem at time t n+. Tus, te proposed metod, wit te viscous diffusion made explicit, allows te vorticity integral conditions to be ful- lled a posteriori, as already pointed out in [6]. In te present formulation te vorticity boundary value is determined in a way tat is very similar to te classical procedure used in te context of nite differences. In fact te vorticity boundary formula used in second-order accurate central differences is obtained by means of a Taylor series expansion as follows ψ ( x) = ψ() + x ψ() x [ ] ( x) ω () + ψ() + O(( x) 3 ). y Tis argument uses te Poisson equation for ψ on te boundary togeter wit te Diriclet and Neumann boundary data for ψ. In some sense, te Taylor expansion above mimics te weak vorticity equation for a weigting function v suc tat v Γ. 5 STABILITY ANALYSIS Te following convergence result is establised in [5]: Γ Teorem Under convenient regularity ypotesis on te solution (ψ, ω) of te continuous problem (), tere is c s(ω) > and c e(t, ν, ψ, Ω) > so tat if δt c s /ν, ten ψ ψ l (H (Ω))+ ω ω l (L (Ω)) c e(δt+ l ). 6 SECOND ORDER BDF SCHEME Te present tecnique is not restricted to rst order; it can be modi ed to obtain ig order accuracy in time. Tis can be done simply by approximating te time derivative by a ig order nite differencing (Crank Nicolson, tree-level backward differencing, etc...) and by extrapolating te terms tat involve ω, accordingly. To illustrate tis possibility we present in te following a second order sceme based on te tree-level backward differencing of te time derivative and using a semi-implicit evaluation of te nonlinear term by means of linear extrapolation in time of te te vorticity. Initialize te sceme by evaluating (ψ, ω) and (ψ, ω). ψ and ω are evaluated from te initial data troug (3) and (). ψ can be obtained by many means; for instance, it can be calculated by using a second order Runge Kutta tecnique; from ψ one evaluates ω easily. Ten, for eac n, carry out te following two steps: Find ψ n+ Ψ, so tat, φ Ψ,, ( (3ψ n+ ψ n + ψ n ), φ ) δt + b(ω n ω n, ψ n+, φ ) = ν ( (ω n ω n ), φ ) + (f n+, φ ẑ), (7) and Find ω n+ W so tat, v W, (ω n+, v ) = ( ψ n+, v ). (8) Tis sceme is second order accurate in time. Its error analysis follows te same ideas as tose tat ave been used to analyze te rst order sceme and a uniform stability result can be found in [5]. 7 NUMERICAT TESTS We ave implemented te tree-level BDF sceme above in two different manners: rst by evaluating te nonlinear term in a fully explicit manner by time extrapolation Explicit ψ-ω metod 6 J.-L. Guermond and L. Quartapelle

4 e 7. e 8 e 8 e 7 e e 7 e 7 e 7 e e 7.. e 7 e 7.7 e Figure. Streamlines of FEM solution of te driven cavity problem for Re = at t = 6.5: fully explicit sceme wit direct solution of te symmetric linear systems. e 8 e. e 7 e e 7. e 8 5e 5 e e 7 e e 7 e 8 e e 7 e 8 Figure. Streamlines of FEM solution of te driven cavity problem for Re = at t = 6.5: semi-implicit sceme wit iterative solution of te linear systems..7 e. e 8 5e 5 e 7 e e e 7. e of bot ψ n+ and ω n+, leading to te weak equation, φ Ψ,, (3 ψ n+ /δt, φ ) = b(ω n ω n, ψ n ψ n, φ ) + g n (φ ), second, by evaluating te nonlinear term by means of a semi-implicit approximation wic gives a nonsymmetric contribution to te elliptic problem for ψ n+, as follows, (3 ψ n+ /δt, φ ) + b(ω n ω n, ψ n+, φ ) = g n (φ ). In bot cases te source term g n (φ ) is de ned by g n (φ ) = ( (ψ n ψ n /)/δt ν (ω n ω n ), φ ). Te rst sceme allows using direct algoritms for solving te two symmetric and time-independent linear sys- tems (stiffness and mass matrix) for ψ n+ and ω n+, respectively. Beside te mass matrix problem, te second sceme as a nonsymmetric system of linear equations to be solved at eac time step. As a consequence, we used te GMRES tecnique wit preconditioning based on incomplete factorization of te (constant) stiffness matrix. Te two algoritms ave been tested and compared by solving te unsteady driven cavity problem using nonuniform meses of 8 linear triangles, rst for Re =. In gures and we report te streamlines at time t = 6.5 calculated by te direct and te iterative FEM sceme using δt =. and δt =.5, respectively. Figures 3 and contains te vorticity distribution at te same time, to be compared wit te spectral solution calculated by a Galerkin spectral metod of Glowinski Pironneau type [] wit δt =. and 5 Legendre polynomials in eac direction, sown in gure 5. Te two FE metods are equally accurate, te sligt differ- Explicit ψ-ω metod 6 J.-L. Guermond and L. Quartapelle

5 ences between tem being caused by te use of different meses. Te second example is te calculation of te same problem for a moderately ig Reynolds number Re =. Te streamlines at time t = 5. provided by te explicit direct FEM sceme wit δt =. are sown in gure 6. Tis solution is in fair agreement wit a nite difference solution obtained on a uniform 5 grid and using a centered -accurate approximation of te fully explicit Jacobian reported in gure Figure 3. Vorticity of FEM solution of te driven cavity problem for Re = at t = 6.5: fully explicit sceme wit direct solution of te symmetric linear systems. 8 CONCLUSIONS In tis paper we ave presented a nite element sceme for solving te time-dependent Navier Stokes equations formulated in terms of te stream function and te vorticity. Te calculation of te stream function and te vorticity are uncoupled owing to an explicit treatment of te viscous diffusion togeter wit a non-standard writing of te evolutionary term in te weak form of te momentum equation. Te explicit treatment of viscous diffusion Explicit ψ-ω metod 6 J.-L. Guermond and L. Quartapelle

6 Figure. Vorticity of FEM solution of te driven cavity problem for Re = at t = 6.5: semi-implicit sceme wit iterative solution of te linear systems..5 Figure 5. Vorticity of Galerkin Legendre spectral solution of te driven cavity problem for Re = at t = 6.5: Glowinski Pironneau uncoupled metod. implies a stability condition of te type: νδt/ c. Tis stability constraint is te price to be paid for te extreme algoritmic simplicity of te proposed uncoupled sceme, especially wen compared to Glowinski Pironneau metod and related tecniques. Wile te stability restriction may be severe for creeping ows, te matter improves for convection dominated ows since te combination of te cell Reynolds number condition for adequate spatial resolution wit te stability condition gives a condition δt c. Te uncoupling strategy proposed in te paper is not limited to time discretizations of low order; a second order accurate sceme based on te tree-level backward difference formula as been presented. REFERENCES [] F. AUTERI AND L. QUARTAPELLE, Galerkin spectral metod for te ζ-ψ equations, submitted for publication, 998. [] W. E AND J.-G. LIU, Vorticity boundary conditions and related issues for nite differences scemes, J. Comput. Pys.,, 996, [3] V. GIRAULT AND P.-A. RAVIART, Finite Element Metods for Navier Stokes Equations, Springer Series in Computational Matematics, 5, Springer-Verlag, 986. [] R. GLOWINSKI AND O. PIRONNEAU, Numerical metods for te rst biarmonic equation and for te twodimensional Stokes problem, SIAM Review,, 979, 67. [5] J.-L. GUERMOND AND L. QUARTAPELLE, Weak approximation of te ψ-ω equations wit explicit viscous diffusion, submitted for publication, 998. [6] M. NAPOLITANO, G. PASCAZIO AND L. QUAR- TAPELLE, A review of vorticity conditions in te solution of te ζ-ψ equations, Computers and Fluids, to appear, 998. [7] R. PEYRET AND T.D. TAYLOR, Computational Metods for Fluid Flow, Springer-Verlag, Berlin, 983. Explicit ψ-ω metod 63 J.-L. Guermond and L. Quartapelle

7 Mon May 8 :: Figure 6. Streamlines of FEM solution of te driven cavity problem for Re = at t = 5.. Figure 7. Streamlines of FDM solution of te driven cavity problem for Re = at t = 5.. Explicit ψ-ω metod 6 J.-L. Guermond and L. Quartapelle