Natural vorticity boundary conditions on solid walls

Transcription

1 Natural vorticity boundary conditions on solid walls Maxim A. Olsanskii Timo Heister Leo G. Rebolz Keit J. Galvin Abstract We derive boundary conditions for te vorticity equation wit solid wall boundaries. Te formulation uses a Diriclet condition for te normal component of vorticity, and Neumann type conditions for te tangential components. In a Galerkin (integral) formulation te tangential condition is natural, i.e. it is enforced by a rigt-and side functional and does not impose a boundary constraint on trial and test spaces. Te functional involves te pressure variable, and we discuss several velocity-vorticity formulations were te proposed condition is appropriate. Several numerical experiments are given tat illustrate te validity of te approac. Introduction Fluid flow vorticity is an important dynamic variable and many penomena can be described in terms of vorticity more readily tan in terms of primitive variables. Vorticity plays a fundamental role in understanding te pysics of laminar, transitional and turbulent flows [7, 36, 3], in matematical analysis of fluid equations [5], and in computational fluid dynamics []. Te vorticity dynamics of incompressible viscous fluid flows is driven by te system of equations w ν w + (u )w (w )u = f () t were u is te fluid velocity in a non-inertial reference frame, w = u is te flow vorticity, ν is te kinematic viscosity coefficient, and f is a vector function of body forces per unit mass. To obtain a closed system, one sould complement () wit equations for u and initial conditions, and if a flow problem is posed in domain wit boundaries, ten boundary conditions sould be prescribed. Commonly, boundary conditions are given in terms of primal variables and stress tensor, rater tan in terms of te vorticity. However, for analysis and for numerical metods based on vorticity equations, it is important to endow () wit boundary conditions on w. Appropriate vorticity boundary conditions ave been a subject of intensive discussion in te literature, especially, in te context of numerical metods for fluid equations. In Department of Matematics, University of Houston, Houston, TX 774 (molsan@mat.u.edu), partially supported by Army Researc Office Grant 6594-MA. Department of Matematical Sciences, Clemson University, Clemson, SC 9634 (eister@clemson.edu), partially supported by te Computational Infrastructure in Geodynamics initiative (CIG), troug te National Science Foundation under Award No. EAR and Te University of California Davis. Department of Matematical Sciences, Clemson University, Clemson, SC 9634 (rebolz@clemson.edu), partially supported by Army Researc Office Grant 6594-MA. Department of Matematical Sciences, Clemson University, Clemson, SC 9634 (kjgalvi@clemson.edu), partially supported by NSF grant DMS593

2 section, we include a brief review of several main approaces. One sould be especially careful wit assigning vorticity boundary conditions on solid walls, i.e. tose parts of te boundary were a fluid is assumed to ave no-slip velocity, since tese regions are responsible for vorticity production and give rise to pysical and numerical boundary layers. An obvious coice of using te vorticity definition w = u for te boundary condition on w is not always optimal wit respect to numerical accuracy. Tis motivated our searc for an alternative way of prescribing boundary conditions on w. Te main result of tis paper is tat on a no-slip boundary, an appropriate vorticity boundary condition is written in terms of a functional corresponding to a certain distribution along te boundary. In PDEs language, suc boundary conditions are called natural. We derive tis condition in section 3. Natural boundary conditions are easy to implement numerically, since tey do not impose boundary constraints on trial and test spaces in a Galerkin metod, and are less prone to produce numerical boundary layers. Our analysis determines tat te functional depends on pressure distribution along te boundary. Te critical role of tangential pressure gradients for boundary vorticity generation is known in te literature and discussed, e.g., in [7, 3]. However, tis relationsip as seemingly not been exploited for devising numerically efficient boundary conditions. In section 3 we also discuss wat new insigt te boundary conditions may give in a possible role of pressure and surface curvature in te vorticity production along solid boundaries. Te remainder of te paper is organized as follows. In section, we give necessary preliminaries and briefly review boundary conditions suggested in te literature to complement te vorticity equations (). Te natural vorticity boundary conditions are derived in section 3. Section 4 discusses several options to close te system of equations by combining () wit te vector Poisson equations for velocity or te momentum equations wit nonlinear terms driven by te Lamb vector. Section 5 presents results of several numerical experiments wic demonstrate te utility and efficiency of te new vorticity boundary conditions for computing incompressible viscous flows. Section 6 collects a few closing remarks. Problem setup and boundary conditions review We consider te flow of an incompressible viscous Newtonian fluid in a bounded domain Ω R 3. In primitive (velocity-pressure) variables, te fluid motion is described by te Navier-Stokes equations u ν u + u u + p = f, t div u =, () u t= = u. We distinguis between te upstream (inflow), downstream (outflow) and no-slip parts of te boundary, Γ in, Γ out, and Γ w ( Ω = Γ in Γ out Γ w ), by te type of boundary conditions imposed on tem. On Γ in we assume a prescribed velocity profile u in and an outflow boundary condition on Γ out, e.g., te vanising normal component of te stress tensor [7]. On te no-slip boundary Γ w we ave u = g on Γ w, wit g n =, (3) were n is an outward normal vector for Γ w and g(x, t) is a tangential velocity of te solid part of boundary. It is common to ave g = for flows past a steady object or cannel flow.

3 For te inflow, one may assume vorticity to be know and set w = u in on Γ in, and letting te normal vorticity derivative vanis is a reasonable outflow boundary condition [3, 6]: ( w)n = on Γ out. Te situation is more delicate wit vorticity boundary conditions on Γ w and several suggestions can be found in te literature. One common coice, see, e.g., [3, 4, 4, 4, 43, 44], is te kinematic condition w = u on Γ w. (4) Note tat opposite to inflow and outflow conditions, te vorticity boundary values on Γ w depend on a generally unknown velocity field. For a computational treatment, tis can be problematic, since numerical differentiation applied to a discrete velocity field on Γ w may reduce te accuracy of te computed vorticity in te wole domain Ω, see [3]. Indeed, a numerical error introduced to vorticity values on Γ w propagates into boundary layers and furter may be convected in te interior of Ω. To mitigate te accuracy reduction and restrictions on w caused by (4), some autors [9, 3], consider a weak numerical enforcement of (4), e.g., by using te Nitsce metod. A variant of (4), suggested in [35, 4, 45], uses only tangential part of te kinematic condition and enforces free divergence of w on Γ w : w n = ( u) n, div w = on Γ w. (5) Te explicit enforcement of te div-free constraint on w along te boundary elps to ensure te vorticity to be solenoidal in Ω. However, it does not resolve accuracy issues related to (4). Anoter class of vorticity boundary conditions are non-local conditions involving integral or integral-differential constraints, see, e.g. [,, 34]. Tis approac uses te Biot-Savart formula to express te velocity from w and furter to find te vorticity diffusive flux on Γ w solving an integral equation. Unlike tis approac, we sall consider local differential boundary conditions on w. If te Navier-Stokes equations are written in terms of vorticity-stream (vector) function, ten boundary conditions following from (4) are written in terms of te stream function rater tan velocity, cf. [9]. 3 Vorticity boundary conditions In tis section, we look for alternative boundary conditions to (4) on solid walls. One vorticity condition easily follows from (3): w n = ( g) n on Γ w. (6) Te rigt and-side of (6) is well-defined, since ( u) n depends only on boundary values of u, as is easy to see from te Kelvin-Stokes teorem (see also Proposition 3. below). Tus, w n is defined only by given boundary values of u, rater tan by te unknown velocity in te fluid domain. 3. Neumann vorticity boundary conditions We need two more vorticity boundary conditions on Γ w. To deduce tem, we rewrite te momentum equation as u + ν w + w u + P = f, t 3

4 were P = u + p is te Bernoulli pressure, p is te kinematic pressure. Taking te tangential component on Γ w and substituting (3) yields: ν( w) n = (f P w g g t ) n = (f P g t ) n g(( g) n) on Γ w. (7) Here we used te identity: (w g) n = g(w n) w(g n) = g(w n) and (6). We note tat te terms on te rigt and side of (7) depend only on boundary values of g and p. Te relation (7) gives two more boundary conditions on w. Since ( u ) n depends only on te boundary values of u, te rigt-and side in (7) can be also rewritten using te kinematic pressure troug ( P ) n = ( p) n + ( g ) n. To simplify te notations, we use te identity g(( g) n) = (( g) g) n, for g satisfying g n =, and denote g = f g t ( g) g ( g ) on Γ w. Te Neumann vorticity boundary conditions now take more compact form: ν( w) n = ( g p) n on Γ w. (8) We obtained Neumann type vorticity boundary conditions (8). Te rigt and side in (8) depends on te gradient of te pressure variable. For f = and g =, te conditions (8) are discussed in [3] as possessing ric pysical and matematical information relevant to vorticity dynamics. In particular, it is noted tat te dynamic boundary conditions for te vorticity and pressure gradient naturally matc te acceleration aderence, wic makes te boundary conditions and compatibility condition merge into one. Tis is argued to reflect te correct pysics, since te acceleration aderence determines te vorticity creation from te boundary. For te furter understanding of conditions (8) assume tat te boundary Γ w is static, tere are no external forces and so g =. Recall tat n is te normal vector pointing from Γ w outward into te fluid domain, ence te vector w n is te boundary vorticity flux, i.e. can be considered as te vorticity production on te solid boundary. Note te identity w n ( w) n = [( w) ( w) T ]n = w n ( w)t n. (9) Denote by H te Weingarten map or te sape operator for te surface Γ w. We need te following properties of H(x), x Γ w : H is a symmetric 3 3 tensor on Γ w, eigenvalues of H are {, κ, κ }, were κ, κ are te principle curvatures of Γ w and te eigenvectors are te corresponding principal directions in wic te surface bends at eac point. It also olds H = Γ n for te surface gradient Γ. Wit te elp of H and recalling w n = on Γ, we can rewrite te last term in (9) as ( w) T (w n) n = (w n) ( n)w = n Hw. n Substituting tis into (9) and (8), gives te expression for te production of te streamwise and spanwise vorticity on te solid wall boundary: ( ) w (w n) n = ν n p + Hw on Γ w. () n n 4

5 On left and side of () one as te total vorticity production on te boundary minus te flux into te normal vorticity component. Hence te left and side corresponds to te boundary flux for te longitudinal and latitudinal vorticity components. Te rigt and side indicates tat te vorticity production depends on te variation of pressure along te boundary and on te sape of te boundary. It is well-known in te literature [7] tat te tangential pressure gradients plays an important role for te generation vorticity on a boundary. Note tat for convex (as viewed from te fluid domain) boundaries, te bilinear form H, is negative definite on planes tangent to Γ w, wile it is positive definite for concave boundaries. Tus, () explicitly sows tat te vorticity production is suppressed along convex boundaries and intensified along concave boundaries. In particular, tis may give a formal matematical explanation to te well-known effects of convex (stabilizing) and concave (destabilizing) sape on turbulent boundary layers [8, 8], and longitudinal vorticity production along concave walls [4]. We sall pursue investigating implications of () on te vorticity generation and boundary layer properties elsewere. Summarizing, conditions (8) possess ric pysical, matematical and geometrical information relevant to vorticity dynamics, and in tis paper we concentrate on employing tem for efficient numerical treatment of incompressible viscous flows. In previous studies, te boundary w p coupling was deemed to cause te basic difficulty of vorticity formulations and an effort was made to bypass tis coupling numerically at te expense of stability/accuracy restrictions in terms of time step and Reynolds numbers. A different point of view and numerical approac is taken in te present article. Several observations play furter a key role and contrast conditions (8) to (4) or (5) and our approac to decoupling strategies: First, Neumann boundary conditions are natural in an integral formulation of te Navier-Stokes equations and so in any Galerkin metod, i.e. te conditions are accounted for by a rigt-and side functional and do not enter te definitions of trial or test spaces. Second, ( p) n depends only on boundary values of p and one can apply integration by parts over Γ w to avoid computing te pressure gradient. Finally, te vorticity equations can be complemented wit velocity-pressure equations driven by te Lamb vector, wic gives te consistent and numerically efficient coupling and provides te pressure for te boundary functional. We explain tese observations below in more detail. From te implementation viewpoint, one may consider involving te pressure gradient along te boundary as a disadvantage of (8). Below we sow tat (8) can be efficiently implemented and computing te pressure derivatives is avoided. 3. Vorticity boundary conditions as a functional Assume u and p are given and consider te integral formulation of te vorticity equation () subject to conditions (6), (8) as well as inflow and outflow boundary conditions : Find w, satisfying w n = ( g) n on Γ w, w = u in on Γ in and ( w Ω t + (u )w (w )u ) v + ν( w) ( v) + ν div w div v dx = ( f) v dx + (( g p) n) v ds for all t >, () Ω Γ w 5

6 for any smoot v suc tat v n = on Γ w and v = on Γ in. Te outflow conditions we used in () read: ( w) n = and div w = on Γ out. Oter outflow boundary conditions are possible, but would lead to extra boundary integral terms in (). To avoid computing pressure gradient over Γ w, we sall rewrite te last term in () using integration by parts on Γ w. For a smoot surface Γ, recall te definition of te surface gradient and divergence: Γ p = p (n p)n and div Γ v = tr( Γ v) wic are te intrinsic surface quantities and do not depend on extensions of a scalar function p and a vector function v off a surface, see, e.g., []. Assume tat Γ is sufficiently smoot and as a boundary Γ wose intrinsic unit outer normal (conormal) is denoted by µ. We will need te following identity for integration by parts over Γ (see [8, ] for more details): ( Γ ) i p ds = κpn i ds + pµ i dl Γ Γ Γ were κ is te surface mean curvature (κ = div Γ n). Tis leads to te following identity: p div Γ v + v Γ p ds = κ(v n)p ds + pv (dl n). () Γ Te definition of te surface gradient immediately yields te identity: Γ ( p) n = ( Γ p) n. Hence assuming Γ w is smoot and tanks to () we manipulate wit te pressure term from (): (( p) n) v ds = (( Γ p) n) v ds = (v n) Γ p ds Γ w Γ w Γ w = div Γ (v n)p ds p(v n) (dl n). (3) Γ w Γ w For te last term in (3), one gets using v n =, n = and vector identities (v n) (dl n) = v dl. Tis identity sows tat (3) is valid also for piecewise smoot surfaces Γ w. Indeed, te formula () can be applied on eac smoot part of Γ w and te contour integrals over sared boundaries of tese parts cancel out. Finally, one can also rewrite te first term on te rigt-and side of (3) using te following simple result. Proposition 3.. Assume v is a vector field defined in a R 3 neigborood of a smoot surface Γ and v n =, ten it olds div Γ (v n) = ( v) n on Γ. (4) In particular, (4) implies tat ( v) n depends only on boundary values of v. Γ 6

7 Proof. Fix any p Γ and consider a sufficiently small neigborood O(p) R 3 of p. Let φ be a signed distance function of Γ in O(p). Ten n = φ is defined in O(p) and coincides wit te normal vector on Γ. Denote by H = φ te Hessian of φ and P = I nn T te normal projector on level sets of φ. Ten it olds (cf, e.g., []): Γ = P on Γ and PH = HP = H in O(p). (5) For a vector n, denote by [n] a 3 3 skew-symmetric matrix, suc tat [n] a = n a for any a R 3. Using tis formalism, we compute div Γ (v n) = tr(p (v n)) = tr (P[n] ( v)) + tr (P[v] H). (6) Using te elementary properties of te trace operation and (5) we get: tr (P[v] H) = tr ([v] HP) = tr ([v] H) =. Te last equality olds since H = H T and [v] = [v] T. Furter, note tat P[n] = [n]. For te first term on te rigt-and side of (6), we ave tr (P[n] ( v)) = tr ([n] ( v)) = 3 v i ([n] ) i,j = ( v) n. x j Summarizing, te integral formulation of te vorticity equations reads: Find w, satisfying w n = ( g) n on Γ w, w = u in on Γ in and ( w Ω t + (u )w (w )u ) v + ν( w) ( v) + ν div w div v dx = ( f) v dx + Ω ( g n) v ds Γ w p( v) n ds + Γ w p v dl Γ w t >, (7) for any smoot v suc tat v n = on Γ w and v = on Γ in. Te Neumann vorticity boundary conditions (8) are accounted by te surface and contour integrals on te rigt-and side of (7) and impose no restrictions on a functional space, were vorticity is sougt. If te pressure distribution is known along Γ w, ten tese integrals are functionals defined for test functions v and tis constitutes te matematically sound problem formulation. Altoug we avoid computing pressure derivatives now, one migt consider bringing back pressure variable into te velocity-vorticity system of equations as a potential downside of te natural vorticity boundary conditions. Tus, te next section discusses several ways of closing te system suc tat te pressure variable is naturally recovered. 4 Velocity-vorticity coupling Tere are several ways to couple equations for velocity and pressure to te vorticity equation. In tis section we discuss tree suc couplings. We start wit recalling te most commonly found in te literature, wic is based on te velocity and pressure Poisson equations. i,j= 7

8 4. Velocity and pressure Poisson equations Typical coupling of velocity to te vorticity equation is done via te relation u = w. (8) Tese relations do not involve pressure, and so tey are not applicable directly for te proposed boundary condition. However, taking divergence of te Navier-Stokes momentum equation provides p = div(u u f). (9) for kinematic pressure or P = div(w u f), () for te Bernoulli pressure. If () or (9) is equipped wit appropriate boundary conditions, ten pressure could easily be incorporated into a time stepping sceme for a velocity-vorticity system of te form (), (8). 4. Lamb vector and Bernoulli pressure Anoter way is to couple vorticity equations to te momentum equations written in te rotations form. Tis gives te coupled system u ν u + w u + P = f, t div u =, () w ν w + (u )w (w )u = f. t In tis system, te Bernoulli pressure is unknown variable and te velocity dynamics is mainly driven by te Lamb vector w u, wic gives te strong coupling of te vorticity and te velocity. It is discussed in [3] ow tis system can be decoupled in an energy stable way in a time-stepping sceme, by extrapolating vorticity in time in te momentum equation (wic linearizes it), ten using te velocity solution in te (now linear) vorticity equation. Furtermore, [5] proves tat for D flows te numerical metod s discrete vorticity and velocity are bot long-time stable in te L and H norms, witout any timestep restriction. 4.3 Lamb vector, rate of deformation tensor and kinematic pressure For certain problems, te use of Bernoulli pressure may lead to increased numerical error in finite element metods were mass conservation is not strongly enforced []. If one is interested in computing lift and drag around an object, solving for Bernoulli pressure can lead to worse accuracy, since a recovered kinematic pressure may be less accurate tan directly computed. In suc cases, it may be advantageous to use a variant of () tat utilizes te vector identity D(u)v = v u ( u) v, () were D(u) = ( u + ( u)t ) represents te rate of deformation tensor. Relation () enables te velocity vorticity coupling by including te Lamb vector in te momentum 8

9 equations witout altering te kinematic pressure. To our knowledge, te identity () was first pointed out in [4] for use in te Navier-Stokes equations. Tis leads to te coupled system u t ν u + w u + D(u)u + p = f, div u =, (3) w ν w + (u )w (w )u = f. t Similar to () te system (3) can be also integrated numerically using time splitting tecniques as sown in te next section. Given tese tree coupling strategies, te autors of tis paper give some preference to tose in () and (3), since tese formulations do not require pressure boundary conditions, directly lead to energy stable finite element discretizations, cf. [3], and admit natural and simple time-stepping strategies. In te numerical examples section, we will test finite element algoritms based on () and (3). 5 Numerical examples We provide four numerical experiments in tis section, wit te goals of ) testing te feasibility of te proposed natural boundary condition for vorticity, and ) testing te accuracy of te velocity-vorticity scemes tat use te natural vorticity boundary conditions. Our results all indicate tat bot te proposed boundary condition and te scemes perform very well, and at least as good as often better tan related scemes tat use only velocity-pressure variables. Te first test is a steady flow wit known analytical solution, and we calculate convergence rates of a metod tat uses te proposed boundary conditions; optimal convergence rates are found. Te second test is for cannel flow past a normal flat plate at Re= and Re=5, and we find te proposed metod to work significantly better tan an analogous metod tat uses velocity-pressure variables only. For te tird test, we consider is time dependent flow around a cylinder, and we compare te solutions of scemes for () and (3). Here, we find for more accurate lift and drag predictions, it is better to use (3) since it uses te usual pressure instead of te Bernoulli pressure. Our final test is for 3D steady flow around a square cylinder at Re=. Because of te singular solution beavior near te sarp edges of te cylinder, getting accurate lift and drag is very callenging, especially if a numerical approac involves iger order flow dynamics variables, suc as vorticity. Hence, tis test requires locally adapted fine meses. We compare solutions of te proposed (steady) velocity-vorticity metod (3) to te standard sceme in velocity-pressure variables on meses up to 5 million dof, and find te proposed metod works well, giving a significantly better lift prediction and sligtly worse drag prediction. 5. Finite element algoritms For te two tests of steady flows, te velocity boundary condition is given as Diriclet on te entire boundary, so we take Γ = Γ w and apply te proposed vorticity boundary condition on te entire boundary. Denoting by τ a mes of Ω, we define te space W g := {w H (Ω) P k (τ ), w n Γ = ( g) n}. 9

10 In particular, te test space is W := Wg for g =. Next, we compute using te following finite element metod: For a given u H (Ω), p L (Γ), and f L (Ω), set g = f ( g) g, and find w W g, satisfying ((u )w, v ) ((w )u, v ) + ν( w, v ) + ν(div w, div v ) = ( f, v ) + ( g n) v ds p( v ) n ds + p v dl v W. (4) Γ Γ For te time dependent problem, tere will be an inflow and outflow wic will be given as analytic functions for velocity (u = g on Γ in and Γ out ). Hence we will take te vorticity as fully Diriclet on tese boundaries, and enforce w = g on Γ in and Γ out. We test finite element implementation of bot () and (3), wic use a natural splitting in te time stepping and tus decouple te velocity-pressure system from te vorticity system by extrapolating te vorticity troug previous timesteps. Hence for tis problem we define te spaces as V g := {v H (Ω) P k (τ ), v Γ = g}, V = Vg W g := {w H (Ω) P k (τ ), w n Γw = ( g) n, w Γin,out = g}. Denoting φ n+ φ n + φ n+ :=, te linearized Crank-Nicolson, finite element discretization of () is ten (at eac timestep): Step : Find (u n+, P n+ ) (V g, Q ) satisfying t (un+ u n, v ) + Step : Find w n+ ( ( 3 wn ) wn ) u n+, v (P n+, v ) + ν( u n+/, v ) = v V ( u n+, q ) = q Q. W g satisfying t (wn+ w n, χ ) + (u n+ w n+, χ ) (w n+ u n+, χ ) + ν( w n+, χ ) + ν( w n+, χ ) = (( f(t n+ )), χ ) + ( g(t n+ ) n) χ ds P n+ ( χ ) n ds + χ dl χ W. Γ w Γ w Γ Γ w P n+ Since te rigt-and side of (5) contains te Bernoulli pressure, te definition of g is adjusted ere to g = f g t ( g) g on Γ w. We will also test a finite element algoritm for te coupled system (3). Tis sceme splits te system in a similar way as for (). Te Step is replaced by Alternate Step : Find (u, p ) (V, Q ) satisfying t (un+ u n, v ) + (wn+ u n+, v ) + (D(u n+ )u n+, v ) (p n+, v ) + ν( u n+/, v ) = (f n+, v ) v V ( u n+, q ) = q Q. (5)

11 Alternate Step is te same as () wit P n+ replaced by te kinematic pressure p n+ and g = f g t ( g) g ( g ) on Γ w. 5. Numerical Experiment : convergence rates Our first numerical experiment is designed to test te accuracy of te proposed boundary condition, by calculating convergence rates of te following finite element metod solution, for a steady problem wit known analytical solution given on Ω = (, ) 3 by u(x, y, z) = {sin(πy), cos(πz), e x } T p(x, y, z) = sin(πx) + cos(πy) + sin(πz). We coose ν =, g = u on Ω, and te forcing f is calculated from te Navier-Stokes momentum equation and te analytical solution. Vorticity approximations are computed using Step, wit t = and taking u n+ as te nodal interpolant of te analytical velocity solution, using Q, Q and Q 3 elements on uniform quadrilateral meses, and te software deal.ii [, 3]. Te calculated vorticity errors and rates are given in Tables -3, and we observe te rates are optimal in bot te L and H norms, for bot coices of elements. For Q 3 elements, we observe a sligtly iger tan optimal rate of convergence in te H norm, but believe tat wit finer meses tis will reduce to tird order. w w rate w w rate /4.44E-.776E+ / E E-.99 /6 9.47E E-. /3.36E-..38E-. Table : Vorticity L and H errors and rates from Q element computations. w w rate w w rate /4.58E E- /8.466E E-.98 /6.835E E-. / E-4 3..E-. Table : Vorticity L and H errors and rates from Q element computations. w w rate w w rate /.5E-.9E- /4.74E E /8.3E E /.34E E Table 3: Vorticity L and H errors and rates from Q 3 element computations.

12 5.3 Numerical Experiment : Flow past a normal flat plate We consider next a numerical experiment for flow past normal flat plate, following [37, 9, 38]. We take as te domain te [ 7, ] [, ] rectangle cannel wit a.5 flat plate placed units into te cannel from te left, and centered top to bottom. Te inflow velocity is taken to be u in =,, and f =. We run tests wit ν = and ν = 5, giving Reynolds numbers of Re = and Re = 5, respectively, based on te eigt of te plate. On te walls and plate, no-slip conditions are enforced for velocity, and no-penetration for vorticity along wit te additional natural boundary conditions derived erein. For te outflow, te zero-traction boundary condition is enforced for velocity, and te omogeneous Neumann condition for vorticity. We note tat, due to te outflow condition, we use only Alternate Step, since it uses usual pressure and tus more easily enforces zero traction (wit te do-noting condition). We compute bot wit te proposed sceme, and for comparison, we also compute using typical scemes for velocity-pressure in primitive variables. In particular, we use te standard Crank-Nicolson linear extrapolation algoritm, wic is given by CNLE-UP: Find (u, p ) (V, Q ) satisfying t (un+ u n, v ) + (( 3 un un ) ) u n+, v (p n+, v ) + ν( u n+/, v ) = (f n+, v ) v V, ( u n+, q ) = q Q. For te Re = 5 simulation, we use an analogous decoupled linearized sceme wit BDF3 timestepping. Te quantities of interest in te simulations are te time averaged drag coefficient and te Stroual number. Te drag coefficients are calculated at eac timestep by te formula, C d (t m ) = LU S ( ν u t S (t m ) ) n y p m n n x were S is te plate boundary, n = n x, n y is te outward normal to S, u ts (t m ) is te tangential velocity, te maximum inlet velocity U =, and L = is te plate lengt. Te drag coefficients are ten averaged over te last periods in te simulation. In all cases, volume integral formulas are used, as tey are believed more accurate []. Te Stroual number was calculated as in [37, 38], using te fast Fourier transform of te transverse velocity at (4.,.) from T=3 to T= to calculate te frequency f, and ten St = fl/u = f 5.3. Re = Te Re = simulation wit te Alternate Step velocity-vorticity (CNLE-VV) sceme was run using (P, P, P ) elements for velocity, pressure and vorticity, and used grad-div stabilization [3]. Tis provided 4,64 velocity dof, 3,73 pressure dof, and,3 vorticity dof, using a Delaunay generated mes wit aspect ratio of approximately 8. A timestep was cosen to be t =., and te simulation was run to te end time of T = ; te flow reaced a statistically steady, periodic-in-time state by around T =. For a fair comparison, we also ran CNLE-UP using te same mes, timestep, and wit te same (P, P ) velocity-pressure element coice. ds,

13 Table 4 sows te time averaged drag coefficients and Stroual numbers from te solutions, along wit reference values from [38]. Comparing to te reference values, we observe tat bot CNLE-VV and CNLE-UP accurately predict te Stroual number, owever te velocity-vorticity metod wit te proposed boundary conditions gives a muc better approximation of te time averaged drag coefficient. Figures sows bot te time averaged vorticities and te T= instantaneous vorticities for CNLE-VV and CNLE-UP. For CNLE-UP, te curl of te velocity was used as te vorticity and for VV, te vorticity variable w was used. We observe tat CNLE-UP vorticity and averaged vorticity appears significantly less resolved tan for CNLE-VV on tis same discretization. Re sceme C d Stroual number CNLE-VV.6.83 CNLE-UP Reference [38].6.83 Table 4: Long-time average drag coefficients and Stroual numbers for te Re= simulations Re=5 For Re = 5 simulations, we found second order in time scemes to be too inaccurate, and so we used BDF3 timestepping, and canged velocity-vorticity sceme and primitive variable sceme accordingly. We will refer to te velocity-vorticity sceme as BDF3-VV, and te primitive variable sceme by BDF3-UP. We again use (P, P ) elements for velocity and pressure, and for vorticity we use P in te BDF3-VV sceme. Here we used a finer Delaunay-generated mes, wic provided 7,37 velocity dof, 899 pressure dof, and 35,635 vorticity dof, and ad an aspect ratio of around 9. A timestep of t =. was used to te end time of T=; a statistically steady, periodic-in-time beavior was reaced by around T=. Stroual numbers were calculated from solutions from T=3 to, and te average drag coefficient was taken by averaging te drag coefficients from te final periods. Table 5 sows te time averaged drag coefficients and Stroual numbers from te solutions, along wit reference values from [38]. Comparing to te reference values, we observe tat bot BDF3-VV and BDF3-UP predict te Stroual number wit good accuracy; te calculated Str number.7 is te closest discrete frequency to.67 (recall an FFT is used from T=3 to T=). We also observe tat BDF3-VV wit te proposed boundary conditions is significantly closer tan BDF3-UP to te reference solution s time averaged drag coefficient. Figures sow bot te time averaged vorticities and te T= instantaneous vorticities for BDF3-VV and BDF3-UP. We observe, as in te Re= case, tat te primitive variable formulation s vorticity and averaged vorticity appears significantly less resolved tan for te velocity-vorticity metod. 5.4 Numerical Experiment 3: Time dependent flow around a cylinder Our next experiment tests te algoritms above based on te coupling () (Step ) and (3) (Alternate Step ), bot using te proposed boundary condition, on te bencmark problem of time dependent flow around a cylinder. Tis test problem is taken from [, 39], 3

14 5 45 CNLE-VV (time averaged) CNLE-UP (time averaged) CNLE-VV (T=) CNLE-UP (T=) Figure : Sown above are plots of te vorticity contours for Re= for CNLE-VV and CNLE-UP, bot time averaged and instantaneous at T=. Re sceme C d Stroual number 5 BDF3-VV BDF3-UP Reference [38] Table 5: Sown above are long-time average drag coefficients and Stroual numbers for te Re=5 simulations. and te domain for te problem is a..4 rectangular cannel wit a cylinder of radius.5 centered at (.,.) (taking te bottom left corner of te rectangle as te origin), see 4

15 BDF3-VV (time averaged) BDF3-UP (time averaged) BDF3-VV (T=) BDF3-UP (T=) 5 5 Figure : Sown above are plots of te vorticity contours for Re=5 for VV and CNLE, bot time averaged and instantaneous at T=. Figure Figure 3: Te domain for te cannel flow around a cylinder numerical experiment. 5

16 Te cylinder, top and bottom of te cannel are prescribed no slip boundary conditions, and te time dependent inflow and outflow profile are u (, y, t) = u (., y, t) = 6 sin(π t/8)y(.4 y),.4 u (, y, t) = u (., y, t) =. Te viscosity is set as ν = 3 and te external force f =. Te Reynolds number of te flow, based on te diameter of te cylinder and on te mean velocity inflow is Re(t). It is known tat as te flow rate increases from time t = to t = 4, two vortices start to develop beind te cylinder. Tey ten separate into te flow, and soon after a vortex street forms wic can be visible troug te final time t = 8. Lift and drag coefficients for fully resolved flows will lie in te reference intervals ([39]) c ref d,max [.93,.97], cref l,max [.47,.49] For te lift and drag to be accurate, te correct prediction of te boundary layer is critical, and tus we believe tis is anoter good test for te proposed vorticity boundary condition. We compute wit te finite element algoritms discussed above, for te systems () (Step - Step ) and (3) (Alternate Step - Step ). A Delaunay triangulation is used as te mes, and it provided 65, 8 velocity dof (and 48,366 pressure dof) using ((P ), P disc, P ) velocity-pressure-vorticity elements. A time step of t =. is used for te timestepping to an endtime of T = 8. Plots of solutions at T=4 and T=6 are sown in Figure 4 for te solution of te sceme for (), and agree well wit te literature [, 4]. Te solution plots of te sceme for (3) are visually indistinguisable from tat of (). Te lift and drag coefficients were calculated using volume integral formulas (see e.g. []) to be: () (Step -Step ): c V V d,max =.877 (3) (Alternate Step -Step ): c V V d,max =.955 cv V l,max =.58 cv V l,max =.47 Hence using te sceme for (3) gave better lift and drag coefficient predictions tat lie in te reference intervals, wile te sceme for () did not. We believe tis difference in accuracies is due to te use of te Bernoulli pressure in (), since Bernoulli pressure is muc more complex (and tus more error prone) tan usual pressure in flows around objects []. Significant error in te Bernoulli pressure would cause error in te vorticity troug te natural boundary condition, particularly near te boundaries, wic in turn would cause velocity error troug te nonlinearity in te momentum equation. 5.5 Numerical Experiment 4: 3D flow around a square cylinder As a final test we compute te 3d flow around a square cylinder bencmark wit Reynolds number. Te geometry setup is given in Figure 5, see [33, 5] for more details and reference values. An important feature of te flow past square cylinder problem is te singularity of te geometry, wic destroys te regularity of te Navier-Stokes solution. Te regularity teory from [6] predicts p / H (Ω) and u / H (Ω) 3. Tis, in particular, implies tat te pressure and vorticity gradients are bot unbounded in te vicinity of te edges. Tis lack of solution smootness makes te correct prediction of drag and lift coefficients 6

17 T = 4 T=6 Figure 4: Velocity vector plot and speed contours at T=4 and T=6 for te D flow around a cylinder using te sceme for (3); te solution plots of () are visually indistinguisable Outflow X Z Inflow.4 Y Figure 5: Setup for te 3d flow around a square cylinder 7

18 a callenging test for a numerical metod based on vorticity as well as for for boundary conditions involving pressure. For te computations we use a code written in deal.ii ([, 3]) using Q-Q Taylor- Hood elements on quadrilaterals. Te starting mes is adaptively refined using a gradient jump error estimator based on te previous velocity. A cut of te meses used in te computations can be seen in Figure 6. Figure 6: Cuts of te adaptively refined meses in te middle of te domain. Te total number of unknowns for velocity and pressure are 53,78, 498,87,,694,79, and 5,85,73, respectively. As te solution for tis Reynolds number is stationary, we solve te coupled vorticity- Navier-Stokes system as a stationary problem (in contrast to te examples before). We first solve te nonlinear velocity-pressure formulation ν( u, v ) + (u u, v ) (p, v ) + γ( u, v ) = (f, v ) v V ( u, q ) = q Q. (6) using a damped Newton iteration until convergence. Note tat we are adding grad-div stabilization wit γ =. to te system to elp wit accuracy and te linear solvers for te saddle point system (see [6]). We ten proceed wit a fixed point iteration for te 8

19 vorticity formulation ν( u, v ) + (w u, v ) + (D(u )u, v ) (p, v ) + γ( u, v ) = (f, v ) v V ( u, q ) = q Q ν( w, χ ) + ν( w, χ ) (w u, χ ) + (u w, χ ) p ( χ ) n ds + p χ dl = ( f,χ ) χ W Γ w Γ w wit te previously computed Navier-Stokes solution as a starting guess. In eac fixed point iteration, we alternate between solving te stationary vorticity system and te stationary and linearized Navier-Stokes formulation. Te iteration is stopped wen te nonlinear residuals reac 6. Te results for lift and drag are given in Table 6. For te vorticity-velocity metod wit te proposed boundary conditions, we observe te convergence of te statistics to te reference values despite a non-smoot and singular beavior of te pressure and vorticity in te vicinity of te upstream cylinder edges, see Figure 7, were te computed solution is visualized. We run experiments wit bot te standard velocity-pressure formulation (6) and te vorticity-velocity formulation (7) wit te proposed vorticity boundary conditions. Wile for bot formulations te statistics of interest converge to reference values, it is interesting tat te vorticity formulation gives muc better results for te drag, wile te lift is better witout using te vorticity form. Note tat te meses are coarse compared to te reference values in [5], wic were computed wit to 3 million unknowns. dofs lift err% drag err% lift err% drag err% reference [5] {w, u, p}-formulation {u, p}-formulation 53, % %.77.% % 497, % % % %,69, % % % % 5,848, % %.76.4% % (7) Table 6: Values for lift and drag for te stationary flow around a 3d square cylinder on a sequence of adaptively refined meses. Underlined numbers are more accurate. Te number of unknowns in column one corresponds to te sum of velocity and pressure unknowns. 6 Conclusions We ave derived and tested new natural vorticity boundary conditions. Te conditions are local and ave been derived directly from te momentum balance for an incompressible fluid witout invoking any furter empirical or ad oc assumptions. We argued tat te devised condition possesses ric pysical and geometrical information relevant to vortex dynamics, and we concentrated on employing tem for te numerical simulation of viscous incompressible flows. Since metods tat solve directly for vorticity are believed to be more accurate near te boundary for vortex dominated flows, using pysically-derived boundary conditions sould elp to improve teir accuracy. Despite te vorticity-pressure coupling, it appears tat te conditions are easy to implement in finite element or oter Galerkin 9

20 Figure 7: Velocity magnitude, pressure, and vorticity magnitude on te midplane around te cylinder (zoomed) and 3d view of streamlines and vorticity magnitude contours around te cylinder. Te singularities in te pressure and vorticity in te corners of te cylinder making tis a very callenging problem.

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