On the accuracy of the rotation form in simulations of the Navier-Stokes equations

Transcription

1 On te accuracy of te rotation form in simulations of te Navier-Stokes equations William Layton 1 Carolina C. Manica Monika Neda 3 Maxim Olsanskii Leo G. Rebolz 5 Abstract Te rotation form of te Navier-Stokes equations nonlinearity is commonly used in ig Reynolds number flow simulations. It was pointed out by a few autors (and not widely known apparently) tat it can also lead to a less accurate approximate solution tan te usual u u form. We give a different explanation of tis effect related to (i) resolution of te Bernoulli pressure, and (ii) te scaling of te coupling between velocity and pressure error wit respect to te Reynolds number. We sow analytically tat (i) te difference between te two nonlinearities is governed by te difference in te resolution of te Bernoulli and kinematic pressures, and (ii) a simple, linear grad-div stabilization ameliorates muc of te bad scaling of te error wit respect to Re. We also give experiments tat sow bad velocity approximation is tied to poor pressure resolution in eiter form. 1 Introduction Te nonlinearity in te Navier-Stokes equations (NSE) can be written in several ways, wic, wile equivalent for te continuous NSE, lead to discretizations wit different algoritmic costs, conserved quantities, and approximation accuracy, e.g., Greso [9] and Gunzburger [1]. Tese forms include te convective form, te skew symmetric form and te rotation form, given respectively by u u, u u + 1 (div u)u, and ( u) u. Te algoritmic advantages and superior conservation properties of te rotation form (summarized in Section.) ave led to it being a very common coice for turbulent flow simulations, see, e.g., C.7. in [5] and []. 1 Department of Matematics, University of Pittsburg, wjl@pitt.edu, ttp:// wjl,partially supported by NSF Grant DMS 5 Departamento de Matemática Pura e Aplicada, Universidade Federal do Rio Grande do Sul, Brazil, carolina.manica@ufrgs.br, ttp:// carolina.manica 3 Department of Matematical Sciences, University of Nevada Las Vegas, Monika.Neda@unlv.edu, ttp://faculty.unlv.edu/neda Department of Mecanics and Matematics, Moscow State M. V. Lomonosov University, Moscow 11999, Russia, Maxim.Olsanskii@mtu-net.ru, ttp:// molsan, partially supported by te RAS program Contemporary problems of teoretical matematics troug te project No and RFBR Grant Department of Matematical Sciences, Clemson University, rebolz@clemson.edu, ttp:// ler 1

2 It is known from Horiuti [1, 15] and Zang [3] tat te rotation form can lead to a less accurate approximate solution wen discretized by commonly used numerical metods. Horiuti [1] and Zang [3] eac give numerical experiments and accompanying analytic arguments suggesting tat te accuracy loss may appen due, respectively, to discretization errors in te near wall regions (Horiuti, for finite difference metods) and to greater aliasing errors (Zang, for spectral metods). We ave noticed te same loss of accuracy in experiments from [1, ] (for finite element metods) and suggest erein a tird possibility tat it is due to a combination of: 1. Te Bernoulli or dynamic pressure P = p + 1 u is generically muc more complex tan te pressure p, and tus. Meses upon wic p is fully resolved are typically under resolved for P, and 3. As te Reynolds number increases, te discrete momentum equation wit eiter form of te nonlinearity magnifies te pressure error s effect upon te velocity error. We will see, for example, tat in te usual formulation Velocity Error Re * Pressure Error, (.13). Tus, interestingly, some of te loss of accuracy, altoug triggered by te nonlinear term, is due to connections between variables already present in te linear Stokes problem. In finite element metods (FEM) te inf-sup condition for stability of te pressure places a strong condition linking velocity and pressure degrees of freedom. Tis condition, wile quite tecnical wen precisely stated, rougly implies tat for lower order approximations te pressure degrees of freedom sould correspond to te velocity degrees of freedom on a mes one step coarser tan te velocity mes, wile for iger order finite elements te polynomial degree of pressure approximations is less tan te polynomial degree of velocity approximations. Tus, in eiter case for velocities u and Bernoulli pressures P wit te same complex structures, as te mes is refined te velocity will be often fully resolved before te Bernoulli pressure is well-resolved, see te experiments in Section 3.3 involving flow around a cylinder. Furter, wen an artificial problem, constructed so te pressure and Bernoulli pressure reverse complexity, is solved te observed error beavior is reversed: te convective form as muc greater error tan te rotation form, Section 3.. Te question of resolution is reminiscent of Horiuti s argument based on truncation errors in boundary layers. For example, even for a simple Prandtl-type, laminar boundary layer, te pressure p will be approximately constant in te near wall region wile te Bernoulli pressure P = p + 1/ u will sare te O(Re 1/ ) boundary layer of te velocity field. Point 3 is possibly related to aliasing errors; interestingly, te aliasing error in using different forms of te nonlinearity is governed by te resolution of te (Bernoulli or kinematic) pressure. Our suggestion of a fix of using grad div stabilization (see Section.3) works in our tests because it addresses point 3 witout requiring extra resolution. Stabilization of grad div type reduces te error in div u, see (.15), and its (bad) scaling wit respect to te Reynolds number. Moreover, since wen div u = te nonlinear terms are equivalent, tis stabilization causes te tree scemes to produce more closely related solutions. Generally speaking, adding te grad div terms to te finite element formulation is not a new idea. Tese terms are part of te streamline-upwinding Petrov-Galerkin metod (SUPG) in [, 1, 9]. However, in practice tese terms are often omitted, and until recently

3 it was not clear if tey are needed for tecnical reasons of te analysis of SUPG type metods only or play an important role in computations. Te role of te grad div stabilization was again empasized in te recent studies of te (stabilized) finite element metods for incompressible flow problems, see e.g. [, 3,, 1, 3], also in conjunction wit te rotation form, see []. We sall tus compare FEM (or oter variational) discretizations of te rotation form of te NSE, given by wit te usual convection form, given by: and te skew-symmetric form, given by: Tese are related by u t u ω + P ν u = f, (1.1) div u =, (1.) u t + u u + p ν u = f, (1.3) div u =, (1.) u t + u u + 1 (div u)u + p ν u = f, (1.5) div u =, (1.) P = p + 1 u and ω = curl u. Finally, we note tat te rotation and convection (or skew-symmetric) forms lead to linear algebra systems wit different numerical properties, wic occur in time-stepping or iterative algoritms for te NSE problem. Wile tere is an extensive literature on solvers for te convection form, see e.g. [7], not so many results are known for te rotation form. However te few available demonstrate some interesting superior properties of te rotation form in tis respect. In [, ] it was sown tat te rotation form enables one to take into account te skew-symmetric part of te matrix in suc a way tat a special pressure Scur complement preconditioner is robust wit respect to all problem parameters and becomes even more effective wen ν. Suc type of result is still missing for te Oseen type systems wit te convective terms. An effective multigrid metod for te velocity subproblem of te linearized Navier-Stokes system in te rotation form was analyzed in [5]. Finally, in [1] te special factorization of te linearized Navier-Stokes system was studied, wic appears to be well suited for te rotation form. Differences between te nonlinearities We now illustrate some differences between te tree different forms of te NSE nonlinearity. First we discuss te Bernoulli pressure, wic is used instead of usual pressure, wit te rotation form of te nonlinearity, and present a bound (based on te velocity part of te Bernoulli pressure) for te rotation form FEM residual in te convective form FEM. Next, we elaborate te difference in conservation laws of te (FEM discretized) nonlinearities. Lastly in tis section, we present a brief description of grad div stabilization, discuss ow it reduces te differences between te nonlinearities, and sow ow its use allows for better scaling of velocity error wit te Reynolds number. 3

4 .1 Rotation form and Bernoulli pressure Te resolution of te Bernoulli pressure (a linear effect) also critically influences te difference between te nonlinearity in te convective and rotation forms. We sow tat it depends upon te resolution of (te zero mean part of) te kinetic energy in te pressure space. Tis is te dominant part of te Bernoulli pressure. To quantify tis dependence, consider te rotation-form-fem for te simplest nonlinear (internal) flow problem, te equilibrium NSE under no-slip boundary conditions. Let U, Q denote te velocity-pressure finite element spaces. Te usual L (Ω) inner product and norm are always denoted by (, ) and. Te velocity-bernoulli pressure approximations u, P satisfy ν( u, v ) (u curl u, v ) + (q, div u ) (P, div v ) = (f, v ) (.1) for all v, q U, Q. If V denotes te usual space of discretely divergence free velocities V := {v U : (q, div v ) =, q Q }, ten te approximate velocity u from (.1) satisfies ν( u, v ) (u curl u, v ) = (f, v ) v V. (.) Similarly, te FEM formulation for te convective form formulation is given by ν( u, v ) + (u u, v ) = (f, v ) v V. (.3) Te natural measure of te distance of te rotation forms approximate velocity from satisfying te convective forms discrete equations is te norm of residual of te former in te latter. Define tis residual r V via te Riesz representation teorem as usual by (r, v ) := (f, v ) [ν( u, v ) (u u, v )], v V. (.) Proposition.1. Let u be te solution of (.1) and let r be its residual in (.3), defined by (.) above. Let v H(div) = v + div v, and Ten, r 1 M = mean( 1 u ) = 1 1 Ω Ω u dx sup v V, div v (r, v ) v inf [ 1 q Q u M] q. Proof. Using te vector identity u curl u + ( 1 u ) = u u gives tat, for any real number M, (and in particular for M = mean( 1 u )), (r, v ) = ( ( 1 u M), v ) = ([ 1 u M] q, v ), q Q. (We ave integrated by parts and used (q, div v ) =, q Q.) Te Caucy-Scwarz inequality and duality implies tat, as claimed, sup v V,div v (r, v ) v inf [ 1 q Q u M] q.

5 . Conservation properties of te nonlinearities Te conservation properties of an algoritm can provide insigt into bot te pysical fidelity and accuracy of its solutions. Fundamental quantities of te NSE suc as energy (E = 1 u ), elicity (H = (u, u)), and in d enstropy (Ens = 1 u ), play critical roles in te organization of a flow s structures. Te NSE olds delicate pysical balances for eac of tese quantities, and tese balances reveal ow eac term of te NSE contributes to teir development. An NSE algoritm enforcing similar balances (e.g. discrete analogs) for energy, and elicity or d enstropy is tus more likely to admit solutions wit similar pysical caracteristics as te true solution. To gain insigt into te balances admitted by an algoritm, conservation laws are typically studied in te periodic case witout external or viscous forces. Altoug tis case is of little practical interest, if an algoritm fails to upold conservation in tis flow setting, it as little ope for predicting correct pysical balances in irregular domains and/or complex boundary conditions. Consider now conservation laws for energy and elicity in Crank-Nicolson FEM scemes for te NSE wit rotation form (1.1)-(1.), convective form (1.3)-(1.), and skew-symmetric form (1.5)-(1.). Te scemes are defined by: given u V, f L (, T ; V ), timestep t >, kinematic viscosity ν >, end time T, find u i V for i = 1,,..., T t satisfying rotation form: 1 t (un+1 u n, v ) +b r (u n+ 1, u n+ 1, v )+ν( u n+ 1, v ) = (f n+ 1, v ) v V (.5) Convective form: 1 t (un+1 u n, v ) + b c (u n+ 1, u n+ 1, v ) + ν( u n+ 1, v ) = (f n+ 1, v ) v V (.) Skew-symmetric: 1 t (un+1 u n, v ) +b s (u n+ 1, u n+ 1, v )+ν( u n+ 1, v ) = (f n+ 1, v ) v V (.7) were b r (u n+ 1, u n+ 1, v ) = (u n+ 1 (curl u n+ 1 ), v ) b c (u n+ 1, u n+ 1, w) = (u n+ 1 u n+ 1, v ) b s (u n+ 1, u n+ 1, w) = (u n+ 1 u n (div un+ 1 )u n+ 1, v ). By coosing v = u n+ 1 in eac sceme and eliminating viscous and external forces, it is revealed tat u n+1 = u n and tus energy is conserved in te rotation (.5) and skew-symmetric (.7) scemes. For te convective form, owever, we do not ave exact energy conservation. Instead (using (q, u n+ 1 ) = in te last step) 1 un+1 = 1 un + t(u n+ 1 u n+ 1, u n+ 1 ) = 1 un + t( 1 1 (un+ ), u n+ 1 ) (.) = 1 un + t inf ([ 1 1 q Q (un+ ) M] q, u n+ 1 ), (.9) were M = mean( 1 1 un+ ) = Ω un+ dx. (.1) Ω 5

6 Exact energy conservation in te convective form sceme (.) tus depends on div u n+ 1 (and te resolution of te key component of te Bernoulli pressure in te pressure space), wic is nonzero since incompressibility is only weakly enforced. Hence it is possible (and well known to be likely) tat tese small errors in te energy balance at eac time step can accumulate and significantly alter te solution. Regarding elicity (3d) and enstropy (d) conservation, by coosing v = P V (curl u ) in te tree scemes, it can be seen tat none of te scemes conserve elicity; indeed all of te tree nonlinearities alters elicity. However, it was sown in [7] tat if te curl in te rotation form nonlinearity is replaced wit te V -projected curl, ten te rotation sceme will conserve elicity. To our knowledge, no suc alterations can be made to (.) or (.7) to maintain pysical treatment of elicity. It is pointed out in [1] tat for finite difference scemes, te rotation form sows superior conservation properties to te convective form in tat rotation form conserves mean momentum, energy, elicity, enstropy and vorticity versus just mean momentum and energy for te convective form..3 Grad div stabilization Te tree forms of te nonlinearity, and tus te tree scemes (.5),(.), and (.7) are equivalent wen div u =. Since tis condition is only weakly enforced, div u may grow large enoug to cause significant differences between te scemes; as our numerical experiments sow, tis is especially true near boundaries for te rotation form. Grad div stabilization can elp to correct tis error for steady, incompressible flow [], and troug our experiments in Sections 3. and 3. we sow tat tis stabilization tecnique is also effective for unsteady flow. To understand better te role of adding te grad div term to te finite element formulation we consider te model case of te Stokes problem: ν u + p = f, and div u = in Ω, u = on Ω. (.11) Given normal velocity-pressure finite element spaces U, Q, satisfying te discrete inf-sup condition, te grad div stabilized FEM for tis problem is: Pick stabilization parameter γ and find u, p U, Q satisfying ν( u, v ) + γ(div u, div v ) (p, div v ) + (q, div u ) = (f, v ), v, q U, Q. (.1) For te case γ = a common argument is to rescale te equations by p = ν 1 p, f = ν 1 f. Tis leads to a parameter-independent Stokes problem wit a new pressure variable and rigt-and side. One can ten use known results for tis Stokes problem (in {u, p}) and transform back to te {u, p} variables. Te first and most basic result in te numerical analysis of te (parameter-independent) rescaled Stokes problem is tat (u u ) C( inf (u v ) + inf p q ). v U q Q Converting back to te original dependent variables gives (u u ) C( inf (u v ) + 1 v U ν inf p q ). (.13) q Q

7 Tis as te interpretation tat: Velocity Error Re * Pressure Error. For example, a furter development of tese estimates give te error bound in te rescaled variables (u u ) + p p C( u + p ). In te original variables tis immediately yields (u u ) + 1 ν p p C( u + 1 p ), (.1) ν wit a constant C tat is independent of ν. Numerical experiments, see [3], sows tat (.1) is sarp. If γ >, (.1) cannot be so rescaled unless γ = ν. Oterwise, for (.1) te following estimate is valid ([3], T.. and.3) ν 1 (u u ) + γ 1 div (u u ) + p p C((ν 1 + γ 1 ) u + p ) (.15) wit a constant C tat is independent of ν and γ. Estimates (.1) and (.15) suggest tat for small enoug ν we ave for γ = : (u u ) ( u + 1 ν p ) for γ = 1 : (u u ) ν ( u + p ). Tus, large pressure gradients compared to te velocity second derivatives may lead to a poor convergence of te finite element velocity if one does not include grad div stabilization. Oterwise, te dependence of (u u ) on ν is muc milder. 3 Tree numerical experiments We consider tree carefully cosen examples tat, we believe, give strong support for te scenario of accuracy loss described in te introduction. We use te software FreeFem++ [13] to run te numerical tests. Te models are discretized wit te Crank-Nicolson metod in time and wit te Taylor-Hood finite elements (continuous piecewise quadratic polynomials for te velocity and linears for te pressure) in space; te nonlinear system is solved by a fixed point iteration. 3.1 Test 1: Poiseuille flow In Ω = (, ) (, 1), a parabolic inflow v(x, t) = and u(x, y, t) = 1 ν y(1 y) (at x = ) is prescribed. No-slip boundary conditions are given at te top and bottom, and te donoting boundary condition is prescribed at te outflow. Te exact solution is well known to be v(x, y) =, u(x, y) = 1 ν y(1 y), p(x, y) = x +, and we take it as our initial condition. Te key conserved quantity in te flow is te flux troug any cross section given by Q(x) = u(x, y)dy = 1 1ν <y<1 Note tat u = (u, v) and p are in te finite element spaces so tat we expect tat discretization of te convective and skew symmetric form of te NSE will ave very small errors (comparable to te errors from numerical integration and solution of te linear and 7

8 nonlinear systems arising). On te oter and, if te rotation form is used te exact solution is v(x, y) =, u(x, y) = 1 y(1 y), ν P (x, y) = p(x, y) + 1 ν y (1 y), is not in te pressure finite element space. Tus, in te rotation form tere will be discretization errors in P tat influence as well te velocity error troug te discrete momentum equation, since P / Q and P = O(ν ). (3.1) To test Poiseuille flow we take te time step t =.1 and number of time steps = 1, so te final time is T = 1. For te flux computations we find tat expressing te nonlinearity in different forms does not affect te true value of te flux for Re = 1. Results are presented in Table 1. Number of degrees of freedom convective form skew-symmetric form rotation form Table 1: Relative flux errors at Re = 1. Next we test te coupling between velocity and pressure errors by computing te error on a fixed mes for Re = 1 and Re = 1 (decreasing ν). We present te results at te final time T = 1 and at te mes level wit number of degrees of freedom being 1 in Table. Reynolds number u u, u u, p p, p p, convective form skew-symmetric form rotation form P P, P P, Table : Velocity and pressure errors for NSE in test 1: various forms and Reynolds numbers. From Table, te convective form of NSE performs best wit respect to te size of te velocity and pressure errors. Te velocity and pressure errors for te skew-symmetric form are bigger tan te corresponding errors for te convective form It is known tat skew-symmetric form of te nonlinear term of NSE imposes difficulties for simulation of Poiseuille flow, [1, 11], wic we also observe ere.

9 We also observe tat wile te velocity errors are smaller for te rotation form compared to te skew-symmetric form, te error in te (Bernoulli) pressure gradient is larger tan te (usual) pressure gradient error of te skew-symmetric form. Note tat for rotation form, te pressure error p p, and te velocity error (u u ), seem to scale like Re 3, and u u, seems to scale like Re. Poor scaling wit Re can be improved in te case of Stokes and rotation form steady NSE wit te use of grad div stabilization, and is our motivation in later test problems to use tis stabilization. 3. Test : Resolution vs. nonlinearity Te relative importance of resolution of pressures vs. nonlinearity can be tested by artificially reversing p and P in Test 1 in a (completely) syntetic test problem. Tus, we take: u(x, y) = 1 y(1 y), v(x, y) = P (x, y) = x + ν so tat p(x, y) = x + 1 ν y (1 y), Tese are inserted in te Navier-Stokes equations to obtain a rigt-and side f = f(x, y, ν): f(x, y; ν) := (, 1 ν (y (1 y) y(1 y) ) ) T Te resolution of p vs. P is exactly reversed from Test 1. We present te error beavior at te final time T = 1 and at te mes level wit number of degrees of freedom being 1 in Table 3. Reynolds number u u, u u, p p, p p, convective form skew-symmetric form rotation form P P, P P, Table 3: Velocity and pressure errors for NSE in test : various forms and Reynolds numebers. Table 3 sows tat discretization errors are present if te solution does not belong to te finite element space. Te computed errors from Test are te mirror image (up to te preset accuracy used for te various linear and nonlinear iterative solvers stopping criteria )of te error beavior in te previous test. Tus it is clear tat, witout grad div stabilization, in te rotation form it is te resolution of te Bernoulli pressure determine te quality of te overall velocity approximation. 9

10 3.3 Test 3: Flow around a cylinder Next we consider te bencmark problem of flow around a circular cylinder offset sligtly in a cannel, from [], see Figure 1. Te primary feature is te von Karman vortex street. To explore resolution vs. solution quality for te two formulations we test at a Reynolds number sligtly above te critical one for vortex sedding and we ave te simple test: Vortex street formed yes or no. Te time dependent inflow and outflow profile are u 1 (, y, t) = u 1 (., y, t) = sin(π t/)y(.1 y).1 u (, y, t) = u (., y, t) =. No slip boundary conditions are prescribed along te top and bottom walls and te initial condition is u(x, y, ) =. Te viscosity is set at ν = 1 3, te external force f =, and te Reynolds number of te flow, based on te diameter of te cylinder and on te mean velocity inflow is Re 1. Te time step is cosen to be t =.5. Figure 1: Cylinder Domain Freefem generated two Delaunay meses for testing tis problem, te finest of wic is able to resolve te problem for te NSE in eiter (rotation or skew symmetric) form. Tese are sown in Figure. Figure : Sown above are two levels of mes refinement provided by Freefem for computing flow around a cylinder. Te meses provide, respectively, 1,55, and 5,7 degrees of freedom for te computations. Te velocity field calculated on mes 1 is sown in Figures 3 and. Note tat 1

11 wit te skew symmetric form te vortex street is well defined already on mes 1 (coarse mes); wit te rotation form te mes 1 simulations fails. wit te rotation form and grad div stabilization, te mes 1 simulation forms a vortex street Figure 3: Sown above is te velocity field at times t =, 3, 5,, 7, for te NSE solved on mes 1 wit te skew-symmetric form of te nonlinearity. Te vortex street forms successfully. Te pressure (and accuracy tereof) is critical for te formation of te vortex street. To test te resolution ypotesis, we move to mes, wic fully resolves bot formulations. Figures 5 and plot p (from skew-symmetric formulation) and P (from rotation formulation), respectively, and from tese plots we see indication tat P contains muc smaller 11

12 Figure : Sown above is te velocity field at times t =, 3, 5,, 7, for te NSE solved on mes 1 wit te rotation form of te nonlinearity. Te vortex street fails to form. transition regions tan p. Te difference can also be seen wen te L norm of p, P are plotted versus time, in Figure 7. Figure sows te effect of te grad div stabilization of te solution computed in mes 1 wit γ = 1 and te rotation form. Witout te stabilization, te rotation form is unable to predict te correct beavior. Wit grad div stabilization, te correct beavior is predicted already on mes 1. 1

13 Figure 5: Sown above is te Bernoulli pressure P at times t =,, 5,, 7, from NSE rotation Form on mes, were a vortex street forms successfully. 3. Test : Cannel flow over a forward and backward facing step Te most distinctive feature of tis common test problem is te formation and detacment of vortices beind te step (a more detailed discussion of tis test problem can be found in Gunzburger [1] and Jon and Liakos [17]). We study te beavior of NSE scemes using te convective form, te skew-symmetric form, te rotation form, and te rotation form wit grad-div stabilization (wit γ =.5). Te simulations are performed on te same domain, wic is mesed wit Delaunay triangulation (provided by Freefem), yielding,59 degree of freedom systems. We set Re = (sligtly above te critical Reynolds number for wic eddies are known to sed), and take timestep t =.5. Results are presented for a parabolic inflow profile, given by u = (u 1, u ) T, wit u 1 = y(1 y)/5, u =. Te no-slip boundary condition is prescribed on te top and bottom 13

14 Figure : Sown above is te usual pressure p at times t =,, 5,, 7, from NSE Skew- Symmetric Form on mes, were a vortex street forms successfully. boundary, as well as on te step. At te outflow te standard do-noting boundary condition is imposed. We conclude tat te NSE wit te convective and skew-symmetric forms of te nonlinearity give te appropriate sedding of eddies beind te step. Te NSE wit te rotation form fails to describe te flow correctly, but te rotation form wit grad div stabilization successfully captures te generation and detacment of eddies. As an interesting but tangential observation, te do-noting outflow boundary condition is not satisfactory for use wit te rotation form wic means tat, until te outflow boundary issue is resolved for te rotation form, for practical purposes one as to use a domain wic is sufficiently large so tat te do-noting boundary condition is applied far enoug from region of interest. As we see in Figure 1, numerical artifacts are seen near 1

15 1 1 grad p, mes grad p, mes 1 grad P, mes grad P, mes t Figure 7: Sown ere is a comparison of te L norm of pressure gradients from simulations of d flow around a cylinder on Meses 1 and. P denoted Bernoulli pressure from te rotation form sceme, and usual pressure from te convective form sceme is denoted by p. It is clear tat P is larger during te development (or lack tereof) of a vortex street. te outflow boundary. Conclusions Altoug te convective, skew-symmetric and rotation forms of te nonlinearity are equivalent in te continuous NSE, in finite element discretizations te rotation form offers better pysical properties (in terms of conservation laws), superior properties for iterative algoritm development, is typically more stable tan te convective form, and is less expensive tan computing te skew-symmetric form. However, using rotation form requires te use of te Bernoulli pressure, wic is generically significantly more complex tan te usual pressure of te convective and skewsymmetric forms. Bernoulli pressure is tus not as easily resolved, wic causes significantly worse results in our bencmark problems for te rotation form sceme. Fortunately, wit te use of grad div stabilization, te inaccuracy in te Bernoulli pressure associated wit using rotation form seems to be localized in te pressure error and ave muc reduced (or even minimal) effect upon te velocity error. 15

16 Figure : Sown above is te velocity field at times t =, 3, 5,, 7, for te NSE solved on mes 1 wit te rotation form of te nonlinearity and wit grad div stabilization. Here te vortex street forms succesfully. 1

17 1 T=1 1 T= T=3 1 T= Figure 9: NSE wit convective form of nonlinearity 1 T=1 1 T= T=3 1 T= Figure 1: NSE wit skew-symmetric form of nonlinearity 1 T=1 1 T= T=3 1 T= Figure 11: NSE wit rotation form of nonlinearity 17

18 1 T=1 1 T= T=3 1 T= Figure 1: NSE wit grad div stabilization for te rotation form 1

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