Numerical Analysis and Scientific Computing Preprint Seria Unconditional long-time stability of a velocity-vorticity metod for te D Navier-Stokes equations T. Heister M.A. Olsanskii L.G. Rebolz Preprint #33 Department of Matematics University of Houston Marc 5
Unconditional long-time stability of a velocity-vorticity metod for te D Navier-Stokes equations Timo Heister Maxim A. Olsanskii Leo G. Rebolz Abstract We prove unconditional long-time stability for a particular velocity-vorticity discretization of te D Navier-Stokes equations. Te sceme begins wit a formulation tat uses te Lamb vector to couple te usual velocity-pressure system to te vorticity dynamics equation, and ten discretizes wit te finite element metod in space and implicit-explicit BDF in time, wit te vorticity equation decoupling at eac time step. We prove te metod s vorticity and velocity are bot long-time stable in te L and H norms, witout any timestep restriction. Moreover, our analysis avoids te use of Gronwall-type estimates, wic leads us to stability bounds wit only polynomial instead of exponential dependence on te Reynolds number. Numerical experiments are given tat demonstrate te effectiveness of te metod. Introduction Te paper addresses long-time stability of numerical metods for te two-dimensional Navier-Stokes system describing te motion of incompressible Newtonian fluids: u t ν u + u u + p = f, div u =, were u = ux, t denotes a velocity vector field, p = px, t is te pressure, and f = fx, t represents given external forcing. Te solution to. is well-known see [5] to be smoot for all time in te periodic setting, tat is, te domain Ω is a D torus T, all functions ave mean zero over te torus, and te forcing term f is smoot. Moreover, te solution of. is longtime stable, in te sense tat te norms u L Ω and u H Ω are bounded uniformly in time for f L R +, L Ω and initial value u H Ω, Ω u =. Te long-time stability is a key property of. if one is interested in simulation of a large time scale penomena or recovering long term statistics, as commonly te case for simulation of flows wit large Reynolds numbers, weater prediction, or climate modeling. Terefore, it is of practical interest to design numerical Department of Matematical Sciences, Clemson University, Clemson, SC 963 eister@clemson.edu, partially supported by te Computational Infrastructure in Geodynamics initiative CIG, troug te National Science Foundation under Award No. EAR-996 and Te University of California Davis. Department of Matematics, University of Houston, Houston TX 77 molsan@mat.u.edu, partially supported by Army Researc Office Grant 659-MA. Department of Matematical Sciences, Clemson University, Clemson, SC 963 rebolz@clemson.edu, partially supported by Army Researc Office Grant 659-MA..
metods for. wic inerit tis important property. It is also interesting to explore to wat extent popular numerical approaces to. are long-time stable. Te topic of long-time stability and error control for numerical metods for te Navier Stokes equations is not new in te literature. Heywood and Rannacer in [, 5] proved uniform in time stability and error estimate in te energy norm for a Crank Nicolson Galerkin metod applied to 3D Navier-Stokes system, assuming te solution of te initial boundary value problem is stable. Simo and Armero in [] examined te long-time stability in te energy norm of several time integration algoritms, including coupled scemes and fractional step/projection metods. More recent studies include te papers [6, 5,, 7, 9]. Te work of Tone and Wirosoetisno [6, 5] proved uniform in time bounds on ut n L Ω and ut n L Ω for implicit Euler and Crank Nicolson metods. Tese bounds are subject to restrictions on time step in terms of ν and a spatial discretization parameter. Badia et al sowed in [] tat u L, ; L Ω for a solution to spatially discretized equations.. First and second order semi-explicit time discretization metods for. written in vorticity stream function formulation were studied by X. Wang and co-workers in [9, 7]. Bot papers consider spectral discretization in space, and prove long-time stability bounds for te enstropy and te H -norm of te vorticity, again all subject to a time step restriction of te form t c Re. Tus, despite progress, te current understanding of te long-time beavior of numerical metods for. is far from being full: only a few studies address uniform in time error estimates for vorticity or velocity gradient, time step restrictions are common in te analyses, and semi-discrete metods are often treated rater tan full discretizations. Moreover, to our knowledge, all proofs of long-time numerical stability bounds for vorticity and te gradient of velocity, invoke a variant of te discrete Gronwall lemma, wic results in te dependence of te bounds on te Reynolds number of te form Oexpc Re or even Oexpc Re. Altoug being time independent, suc bounds are not very practical for iger Reynolds number flows; see [7] for a discussion and an effort to improve numerical stability and error estimates dependence on Re number, but only locally it time. In tis paper, we prove unconditional long-time stability of a fully discrete numerical metod for.: For f L, ; H Ω we prove uniform in time estimates for te kinematic energy, enstropy, as well as te L norms of velocity gradient and vorticity gradient of a discrete system. A finite element metod is used for te spatial discretization, and bot first and second order time stepping semi-implicit linear at eac time step scemes are studied. Te stability bounds are unconditional, i.e. absolutely no time step restrictions are imposed. Furtermore, our analysis does not rely on any Gronwall type estimate, wic allows us to avoid exponential dependence of stability bounds on te Reynolds number. In te present analysis, te dependence is polynomial. Our analysis reveals tat te polynomials degree can be significantly lowered at te expense of logaritmic dependence on te spatial mes size. Te results of te paper systematically exploit te relationsip between te vorticity and velocity of te Navier-Stokes system by considering te vorticity dynamics equation and writing te inertia in te momentum equation in te form of Lamb vector. For w = u and P = u + p, we reformulate. as: u ν u + w u + P = f, t div u =,. w ν w + u w = f. t
Vorticity plays a fundamental role in fluid dynamics, and studying properties of. troug te vorticity equation is a well establised approac in te Navier-Stokes teory, see, e.g., [9, 6]. It is also not uncommon in numerical analysis to design numerical metods based on te vorticity equation, e.g., [8, ]. For numerical metods, standard closures for te vorticity equations are obtained eiter in vorticity stream function variables or wit te elp of te vector Poisson equation, u = w. However, recent papers [, 8] ave demonstrated numerical advantages of complementing te vorticity equation wit te velocity dynamic equation as in.. Tus,. will be te departure point in te present analysis. Te rest of te paper is organized as follows. Section gaters necessary definitions and preliminary results for te analysis tat follows. In Section 3, we introduce a first order time stepping metod and prove its long-time stability wit respect to te velocity and vorticity H norms. Section introduces a second order metod based on BDF time discretization. We extend te long-time stability results for tis metod by taking care of some extra tecnical details. Since te numerical sceme is non-standard, we also provide wit our analysis a series of numerical experiments for a D flow past a bluff object. Te results of te experiments are presented in Section 5., and tey illustrate te long-time stability and te performance of te metod. We finis te introduction wit te following remark. Most of our stability analysis is restricted to te D case and, due to te current lack of understanding of te long time beavior of 3D Navier-Stokes solutions, we cannot say to wat an extend te results remain valid in 3D. However, te numerical approac studied ere as a straigtforward extension to 3D, and relying on a past experience, we believe tat numerical metods wic are pysically consistent and computationally efficient for D problems are commonly found to be also advantageous for solving 3D Navier-Stokes equations. Notation and Preliminaries We consider a domain Ω =, π R, and we restrict tis study to te case of periodic boundary conditions. We note tat our stability analysis also olds for te case of full Diriclet velocity and vorticity boundary conditions. We use te notation, and for te L Ω inner product and norm, respectively. All oter norms will be clearly labeled wit subscripts. Te natural velocity and pressure spaces in te periodic setting for te Navier-Stokes equations are X := H# Ω = {v Hloc R, v is π-periodic in eac direction, v dx = }, Ω Q := L # Ω = {q L loc R, q is π-periodic in eac direction, q dx = }. In two dimensions, vorticity is considered as a scalar, and we define vorticity space as Y := H# Ω = {v H loc R, v is π-periodic in eac direction, v dx = }. For te discrete setting, we assume τ is a regular, conforming triangulation of Ω wic is compatible wit periodic boundary conditions. Let X, Q X, Q be inf-sup stable velocitypressure finite element spaces, Y Y be te discrete vorticity space, all defined as piecewise polynomials on τ. 3 Ω Ω
Te discretely divergence-free subspace will be denoted by V := {v X, v, q = q Q }. Te dual space of V is denoted by V wit norm V. We will utilize in our analysis discrete analogues of te Laplacian operator. Define to be te discrete Laplacian operator on Y : Given φ H Ω, φ Y satisfies φ, v = φ, v v Y. Define A to be a discretely divergence-free Laplace operator, often referred to as a Stokes operator by: Given φ H Ω, A φ V satisfies or equivalently, A φ, v = φ, v v V, A φ, v λ, v + A φ, q = φ, v v, q X, Q. Te Poincare inequality will be used eavily trougout: tere exists λ, dependent only on Ω, satisfying φ λ φ φ X, An immediate consequence on te Poincare inequality and te definition of discrete Stokes and Laplace operators is tat te following bounds old v λ A v v V, z λ z z Y. We recall te following discrete Agmon inequalities, wic are also consequences of discrete Gagliardo- Nirenberg estimates, see [3] p.98: v L C v / A v / v V,. z L C z / z / z Y,,. were C is independent of. Te discrete Sobolev inequality proven in [], φ L C φ φ φ X,.3 again wit C independent of, allows us to prove te following lemma. Lemma.. For every z Y, tere exists a constant C, independent of, satisfying Proof. By Hölder s inequality, and tus using.3 provides te bound z L 3 C z /3 z /3 z Y.. z 3 L 3 z z L, z 3 L 3 z z. Since z = z, z = z, z z z, te estimate becomes z 3 L 3 z z. Taking cube roots of bot sides completes te proof.
Define te skew-symmetric trilinear operator b : X Y Y R by b u, w, χ = u w, χ + uw, χ. We will exploit te property tat b u, w, w = in our analysis of te vorticity equation. 3 Backward Euler We first consider long-time stability of te velocity-vorticity sceme wit finite element spatial discretization and backward Euler temporal discretization. Te algoritm decouples te vorticity equation by using a first order approximation of te vorticity in te momentum equation, and reads as follows. Algoritm 3.. Given te forcing f and initial velocity u, set u to be te interpolant of u, and w te interpolant of te curl of u. Select a timestep t >, and for n=,,,... Step : Find u, p X, Q satisfy for every v, q X, Q, u t u n, v + w n u, v p, v + ν u, v = f, v. 3. u, q =, 3. Step : Find w Y satisfy for every χ Y, w t w n, χ + b u, w, χ + ν w, χ = f, χ. 3.3 We will prove long-time L and H stability of bot te velocity and te vorticity. We begin wit te L results. Teorem 3. Long-time L stability of velocity and vorticity. Suppose f L, ; L Ω, and u H Ω. Denote := + νλ t. For any t >, we ave tat solutions of Algoritm 3. satisfy for every positive integer n, u n + ν w n + ν n k= n k= n k u k+ n k w k+ n u + n w + ν λ f L, ;V =: C, 3. ν λ f L, ;L Ω =: C, 3.5 Remark 3.. Te constants C and C are independent of n and terefore old for arbitrarily large n. Tese bounds can be considered as dependent only on te data since time step sizes are inerently bounded above, and moreover, for sufficiently large n te bounds are independent of te initial condition. Proof. Take v = tu, q = p, and χ = tw, wic vanises te nonlinear and pressure terms, and leaves u u n + u u n + tν u = tf, u, w w n + w w n + tν w = t f, w. 5
We majorize te forcing terms after integrating by parts in te vorticity equation forcing term, applying Young s inequality, and dropping positive terms on te left and sides to get u u n + 3 ν t u ν t f V, w w n + 3 ν t w ν t f. From ere, te velocity and vorticity estimates follow identically, except tat te norm on te forcing term is different, and tus we restrict te remainder of te proof to only te velocity. Applying te Poincare inequality to lower bound te viscous term yields + νλ t u + ν t u u n + ν t f V. Now fix an integer N > and divide te above inequality by N n to obtain N n u + N n ν t u Summing up for n =,..., N and reducing, we get u N + ν N n= N n u N n u n + N n ν t f V. N N u + ν t f L, ;V n= N u + ν t f L, ;V. N n Substituting for proves te velocity result. Applying te same steps for vorticity produces estimate 3.5, wic finises te proof of te teorem. Teorem 3. Long-time H stability of velocity. Suppose f L, ; L Ω, and u H Ω. Denote := + νλ t. For any t >, te solutions of Algoritm 3. satisfy for every positive integer n, n u n u + ν f L, ;L + Cν 3 CC νλ =: C. 3.6 and n u n u + ν λ f L, ;L + C ln ν CC =: C. 3.7 were C is a generic constant, wic depends on Sobolev s embedding inequalities optimal constants and constants from Agmon s type inequalities... Remark 3.. Te teorem above proves tat te long-time velocity solution is bounded in te H norm only by te problem data, and similar to te L bound, it is independent of te initial condition wen n is sufficiently large. Wit respect to te dependence on Re, te estimate 3.6 gives u n ORe5, wile estimate 3.7 gives u n O ln Re 3. 6
Proof. Take v = ta u in 3. to obtain u n + 3 ν t A u ν t f + t w n u, A u. u For te last term on te rigt-and side, we majorize it first using Holder s inequality, te discrete Agmon inequality., Young s inequality, and Teorem 3. to find w n u, A u w n u L A u C w n u / A u 3/ Combining tese last two inequalities produces Cν 3 w n u + ν A u Cν 3 C C + ν A u. u u n + ν t A u ν t f + C tν 3 C C, and tanks to Poincare, we obtain + νλ t u u n + t ν f L, ;L + Cν 3 CC. Recalling te notation = + νλ t, tis relation can be written as u un + ν t f L, ;L + Cν 3 CC. 3.8 Recursive substitution and an estimate for te partial sum of a geometric progression lead us to 3.6. Alternatively, we can employ te finite element inverse inequality u L Ω C ln u, valid in D [3], and estimate te nonlinear terms in te different way: w n u, A u Similar arguments tat produced 3.8 give w n u L A u C ln w n u A u C ln ν C u + ν A u. u un + t ν f L, ;L + C ln ν C u. Doing recursive substitution and employing 3. to estimate te resulting sum k= k n u k leads to 3.7. Teorem 3.3 Long-time H stability of vorticity. Suppose f L, ; H Ω, and u H Ω. Let := + νλ t. For any t >, solutions of Algoritm 3. satisfy for every positive integer n, n w n w + ν f L, ;H Ω + ν 5 CC 6 + Cν 3 CC νλ, 3.9 7
and w n n w + ν λ f L, ;H Ω + C ln ν C C. 3. Remark 3.3. Te teorem above proves tat te long-time vorticity solution is bounded in te H norm only by te problem data, and similar to te L bound, it is independent of te initial condition wen n is sufficiently large. Wit respect to te dependence on Re, te estimate 3.9 gives u n ORe9, wile estimate 3.7 gives u n O ln 3 Re 5. Proof. Take χ = t w in 3.3, and majorize te forcing term using Caucy-Scwarz and Young s inequalities to obtain w w n + 3 ν t w ν t f H + t b u, w, w. We bound te nonlinear term using Holder, Sobolev embeddings, discrete Agmon. and discrete Sobolev inequality., and Teorems 3. and 3. to reveal b u, w, w u w u L 6 w, w L 3 w + u w, w + u w L w CC w /3 w 5/3 + CC w / w 3/ CC C /3 w 5/3 + CC C / w 3/. Te generalized Young s inequality now provides te bound b u, w, w Cν 5 C 6 C + Cν 3 C C + ν w. Combining te estimates above yields w + ν t w w n + C t ν f H + ν 5 C 6 C + Cν 3 C C, and after applying Poincare we get + λ ν t w w n + C t ν f H + ν 5 C 6 C + Cν 3 C C. Te remainder of te proof of 3.9 follows analogous to te H case for velocity. Alternatively, we may bound te nonlinear terms as follows: b u, w, w u w, w u L w w + u w, w + u w L w C ln u w w C ln C w w Cν ln C w + ν w. To complete te proof of 3. we proceed as above and employ estimate 3.5 for te weigted sum of w norms. 8
Second-order metod We consider next a velocity-vorticity sceme wit BDF timestepping. Te sceme decouples te update of velocity and vorticity on eac time step. Similar to te backward Euler case, we sall prove tat te velocity and vorticity are bot unconditionally long-time stable in bot te L and H norms, and te scalings of te stability estimates wit Re are te same as tose from te backward Euler analysis. However, te analysis is somewat more tecnical ere, and a special norm is used to andle te time derivative terms. Algoritm.. Given te forcing f and initial velocity u, set u = u to be te interpolant of u, and w = w te interpolant of te curl of u. Select a timestep t >, and for n=,,,... Step : Find u, p X, Q satisfy for every v, q X, Q, 3u t u n + un Step : Find w Y satisfy for every χ Y,, v + w n w n u, v p, v + ν u, v = f, v,. u, q =.. 3w t w n + wn, v + b u, w, χ + ν w, χ = f, χ. For te matrix G := / 5/ we introduce te G-norm χ G = χ, Gχ, χ is vector valued. Te G-norm is widely used in BDF analysis, see e.g. [, ]. Te following property of te G-norm is well-known []: setting χ = [v, v ] T and χ = [v, v ] T, one gets 3 v v + v, v =, χ G χ v G v + v +. It is also known tat te G norm is equivalent to te L Ω norm in te sense of tere existing C l and C u suc tat C l χ G χ C u χ G. Use of te G-norm and tis norm equivalence will allow for a smooter analysis. We begin our analysis wit te long-time L stability of velocity and vorticity. Teorem. Long-time L stability of velocity and vorticity. Let f L, ; V and u H Ω. Ten for any t >, solutions of Algoritm. satisfy for every positive integer n, C l u n + u n + ν t n un + + max t, C u u + ν t u νc l ν f L, ;V =: C..3 9
If additionally f L, ; L Ω and w H Ω, ten for any t >, solutions of Algoritm. satisfy for every positive integer n, C l w n + w n + ν t n wn + + max t, C u w + ν t w νc l ν f L, ;L Ω =: C 5.. Remark.. A more tecnical analysis can be made, similar to te backward Euler case, tat includes te terms ν n k= k n u k+ and ν n k= k n w k+ on te left and sides of.3 and., respectively. Proof. Coose v = tu in., wic vanises te nonlinear and pressure terms, and ten upper bound te forcing term just as in te backward Euler case to get χ G χ n G + u u n + un + ν t u ν t f V,.5 were χ = [u n, u ] T and χ n = [u n, u n ]T. Dropping te second term on te left-and side, and adding ν t un to bot sides produces χ G + ν t u + ν t u + u n + ν t χ n G + ν t un u + ν t f L, ;V..6 Using te Poincare inequality and ten te equivalence of te G-norm wit te L norm, we ave tat ν t u + u n νλ t and tus setting := min{/, ν tc l }, it olds tat ν t u + u n + ν t Combining.7 and.6 yields + u + u n = ν t χ ν tc l χ G, u χ G + ν t u wic immediately implies tat χ n G + ν t n un + χ G + ν t u..7 χ n G + ν t un + ν t f L, ;V, χ G + ν t u + + +... + + n ν t f L, ;V..8
Since >, and tus n + +... + = + + + + + + χ n G + ν t un n χ G + ν t + u + max{ t, = = max{, ν tcl }, νc l }ν f L, ;V..9 Now using te equivalence of norms for te G norm and L norm of χ completes te velocity proof. Te proof for vorticity follows identically, modulo a iger order norm on te forcing, after taking te text function to be w. We prove next te unconditional long-time H stability of velocity. Teorem. Long-time H stability of velocity. Let f L, ; L Ω, u H Ω, and set := min{/, ν tc l }. Ten for any t >, solutions of Algoritm. satisfy for every positive integer n, C l u n + u n + ν t A u n + + max t, νc l n C u u + ν t A u ν f L, ;L Ω + Cν 3 C 5C =: C 6.. Remark.. Similar to te backward Euler case, te long-time H stability bound for velocity gives u n ORe5. If we instead bounded te nonlinear term as in te backward Euler case via t w n wn u, A u C ln ν C5C, ten we can get instead u n O ln / Re 3. Proof. Coose v = ta u in., wic vanises te pressure terms, and ten upper bound te forcing term to get χ G χ n G + u u n + un + ν t A u ν t f t w n wn u, A u,. were χ = [A / un, A/ u ] T and χ n = [A / un, A / un ]T. Te last term on te rigt and side is estimated using te same tecnique as in te backward Euler case from Section 3, and ten applying te L stability estimates wic is from Teorem. in tis case: t w n wn u, A u C tν 3 w n + w n u + ν t A u C tν 3 C 5C + ν t A u..
Combining tis wit., dropping te second term on te left-and side, and adding ν t A u n to bot sides produces χ G + ν t A u + ν t A u χ n G + ν t A u n + t + A u n + ν t A u ν f L, ;L Ω + Cν 3 C5C..3 From ere, setting := min{/, ν tc l } and taking analogous steps as in te proof of te long-time L estimate starting from.6 provides us wit χ n G + ν t A u n + n χ G + ν t A u + max{ t, νc l } ν f L, ;L Ω + Cν 3 C5C.. Finally, applying te norm equivalence for te G-norm finises te proof. We can now prove te unconditional long-time H stability of te vorticity. Teorem.3 Long-time H stability of vorticity. Let f L, ; H Ω, u H 3 Ω, and set := min{/, ν tc l }. Ten for any t >, solutions of Algoritm. satisfy for every positive integer n, C l w n + w n + max t, + ν t w n νc l n C u w + + ν t ν f L, ;H Ω + Cν 5 C 6 6C 5 + ν 3 C 6C 5 w =: C 7..5 Remark.3. Similar to te backward Euler case, we find tat w n ORe9. However, different estimates of te nonlinear terms i.e., using an inverse inequality as in te backward Euler case can be used to find w n O ln 3/ Re 5. Proof. Begin by coosing χ = t w to get χ G χ n G + w w n + wn + ν t w ν t f + tb u, w, w,.6 were χ = [ / w n, / w ] T and χ n = [ / w n, / w n]t. Upper bounding te nonlinear term exactly as in te backward Euler case, and ten using te long-time estimates proven above for te BDF sceme gives tb u, w, w Cν 5 C 6 6C 5 + Cν 3 C 6C 5 + ν t w,
and tus using tis and dropping te second term on te left side of.6 yields χ G χ n G + ν t w ν t f + C t ν 5 C 6 6C 5 + ν 3 C 6C 5..7 From ere, te same tecniques as for te long-time H stability of velocity can be used to complete te proof, modulo a iger norm on te forcing term. 5 Numerical Experiments We run several numerical experiments in order to test te long-time stability of Algoritm., wic is te BDF timestepping algoritm for te proposed velocity-vorticity metod. However, as our interest is in practical applications, we do not consider a test problem wit periodic boundary conditions; instead, we consider D cannel flow past a flat plate, wic uses a Diriclet velocity inflow, no-slip velocity on te walls, and a zero-traction outflow condition. Tus we must appropriately modify Algoritm. so tat pysical boundary conditions for te velocity and vorticity can be applied. As a numerical illustration of te long term numerical stability, we compute te flow past a normal flat plate, following [, 3]. We take as te domain Ω = [ 7, ] [, ], wit a ole of size.5 representing te flat plate removed from 7 units into te cannel from te left, vertically centered. Te inflow velocity is u in =, T, and no-slip velocity is enforced on te walls and plate. Direct numerical simulations for tis experiment are done for various Reynolds numbers Re, wic can be considered ere as Re = ν, since te lengt of te plate is, and te inflow velocity as average magnitude. Tis is relatively simple, but interesting problem, wic resembles te flow past oter bluff objects. Te flow undergoes a first Hopf bifurcation from steady to unsteady at a relatively low Reynolds numbers between 3 and 35 [] and a second transition, also known as spatial transition from two-dimensional to tree-dimensional, occurs around Re= []. We will test te velocity-vorticity algoritm and its long-time stability for Re= and Re=5. Te matematical formulation of te problem as a constant in time non-omogeneous inflow boundary condition and zero source term. We deem tis setting somewat similar to te one analyzed in te paper periodic boundary conditions and L, ; H Ω-bounded rigt and side, but more practically relevant. 5. Velocity-vorticity formulation wit boundary conditions Denote te domain by Ω, wit boundary Ω = Γ Γ Γ w and Γ being te outflow boundary, Γ te inflow boundary, and Γ w te walls and plate. Denote by τ a regular, conforming triangulation of Ω. Te trial and test spaces for velocity functions are defined by X := {v C Ω P τ, v Γ Γ w = }, X g := {v C Ω P τ, v Γ Γ w = g}, wit g =, T at te inflow, g =, T on te walls, and wit P τ denoting te space of globally continuous functions wic are quadratic on eac triangle. Te discrete pressure space is 3
taken to be Q = {q C Ω P τ }, and te zero traction boundary condition will be enforced weakly in te formulation. Note tat X, Q is te Taylor-Hood velocity-pressure element, wic is known to be inf-sup stable [3]. Te vorticity trial and test spaces are equal, since we take te vorticity at te inflow to be. Te outflow condition for vorticity is a omogeneous Neumann condition, wic is enforced weakly by te formulation. Te natural vorticity boundary condition on Γ w resulting in te presence of te following terms in te finite element formulation, cf. [7], ν w n χ ds = p χ n ds p χ dl χ W. Γ w Γ w G w Te term is added to te formulation wit te known pressure from Step. Tus te vorticity space is W := {w C Ω P τ, w Γ = }. A second modification is made to te algoritm to avoid using te Bernoulli pressure, since tere is an outflow boundary. Here, we use te identity from [], u u + p + u = u u + p + Duu, were Du = u + u T is te rate of deformation tensor. Since tere is no forcing in tis test problem, we set f =, and tus now Steps and of Algoritm. can now be written as tey are computed: Step : Find u, p X g, Q satisfying t 3u + Du Step : Find w u n + un, v + w n w n u, v u n un, v p, v + ν u, v = v X u, q = q Q. W satisfying t 3w w n + wn, χ + u w, χ + ν w, χ = Γ w p χ n ds + G w p χ dl χ W. 5. We note tat since globally continuous pressure elements are used, te rigt and side of 5. can be equivalently written as p χ n ds + Γ w p χ dl = G w p n χ ds. Γ w
5. Cannel flow past a flat plate at Re= and Re=5 Te BDF velocity-vorticity sceme was computed for bot Re= and Re=5 ν=re, using 3 Delaunay generated triangular meses wic provided 7959 total degrees of freedom dof, 65 dof, and 5955 dof wit te P, P, P velocity-pressure-vorticity elements. Te simulations started te flow from rest u =, and were run to an endtime T=. For eac mes, several timestep coices were made, starting wit t=., and ten cutting t in alf until convergence i.e. successive solutions statistics matced. For bot Re= and Re=5, te smallest t was.. Quantities of interest for tis problem is te long-time average of te drag coefficient C d, and te Stroual number. Te Stroual number was calculated as in [, 3], using te fast Fourier transform of te transverse velocity at.,. from T= to T=. Te drag coefficients are defined at eac t n to be C d t m = ρν u t S t m n y p m n n x ds, ρlu max S were S is te plate, n = n x, n y is te outward normal vector to S pointing into te domain, u ts t m is te tangential velocity of u m, te density ρ =, te max velocity at te inlet U max =, and L = is te lengt of te plate. Te integral is calculated by transforming it into a global integral, wic is believed to be more accurate [6]. Te results for time averaged C d and te Stroual numbers from te simulations for eac Re, and for eac mes wit t =., are sown in Table, along wit reference values taken from [3]. We observe tat te 6K dof mes and te 59K dof meses agree well wit te reference values at Re= and Re=5. It appears we ave acieved or are close to grid-convergence, and we note tat for te Stroual number, since te FFT was used wit 8, timesteps,.77 was te closest discrete frequency value to.7, and.89 was te next biggest discrete value compared to.83. We also plot te time-averaged vorticity in Figure, and instantaneous velocity as speed contours in Figure ; bot plots matc te reference plots given in [3]. Metod Mes Re C d Stroual number Vel-Vort 78K dof.8.95 Vel-Vort 6K dof.59.89 Vel-Vort 59K dof.58.89 Saa [3].6.83 Vel-Vort 78K dof 5.57.89 Vel-Vort 6K dof 5.6.77 Vel-Vort 59K dof 5.59.77 Saa [3] 5.55.7 Table : Sown above are Stroual numbers and long-time average drag coefficients for solutions on varying meses, for Re= and Re=5. Reference values are also given for comparison. Also of interest is te stability of computed solutions in te u n L, un H, wn L, wn H norms versus time t n, since we proved in Section tat tese norms are all long-time stable at least, in te periodic setting, independent of te timestep t and mes widt. Plots of tese 5
norms versus time are sown for Re= in Figure 3 and for Re=5 in Figure for varying timesteps. Eac norm appears to be long-time stable. Moreover, we do not observe te very large scaling of any of te norms wit Re. Altoug w n O5 is an order of magnitude larger tan w n, it is still a very reasonable size and nowere near ORe9 or even O ln 3 Re 5. Re= 5 5 5 Re=5 5 5 5 5 5 Figure : Sown above are plots of te time-averaged vorticity contours. Re=.5.5 5 5 Re=5.5.5 5 5 Figure : Sown above are plots of te speed contours of te velocity solutions at T=. 6
u n 5.5 3.5 3 t=. t=.5 t=. t=. t=..5 6 8 6 8 t n u n 36 3 3 3 8 t=. t=.5 t=. t=. t=. 6 6 8 6 8 t n w n 3 t=. t=.5 t=. t=. t=. w n 5 3 t=. t=.5 t=. t=. t=. 6 8 6 8 t n 6 8 6 8 t n Figure 3: Sown above are plots of te Re= solution norms versus time, found using Mes 3 te finest mes. u n.5 3.5 3 t=. t=. t=. u n 38 36 3 3 3 t=. t=. t=..5 8 6 8 6 8 t n 6 6 8 6 8 t n 5 t=. t=. t=. 5 t=. t=. t=. w n 3 w n 3 6 8 6 8 t n 6 8 6 8 t n Figure : Sown above are plots of te Re=5 solution norms versus time, found using Mes 3 te finest mes. 7
6 Conclusions and Future Directions We ave proven unconditional long-time stability of a sceme based on a velocity-vorticity formulation, and a finite-element-in-space BDF-in-time IMEX discretization for te D Navier-Stokes equations. Long-time stability was proven in bot te L and H norms for bot velocity and vorticity, and te estimates old for any t >. Te sceme is non-standard, and so we tested it on a bencmark problem on flow past a flat plate; it performed very well. It would be interesting to study Algoritm., and variations tereof, for 3D flows. Te difference in 3D is tat te vortex stretcing term w u appears in te vorticity equation. Since te D algoritm is proven erein to be unconditionally long-time stable, any instability in te 3D algoritm can be immediately attributed to te vortex stretcing term and/or its numerical treatment. Isolating tis beavior may give insigt into better stabilization metods for iger Reynolds number flows in 3D. References [] S. Badia, R. Codina, and J. V. Gutiérrez-Santacreu. Long-term stability estimates and existence of a global attractor in a finite element approximation of te Navier-Stokes equations wit numerical subgrid scale modeling. SIAM Journal on Numerical Analysis, 83:3 37,. [] R. Bensow and M. Larson. Residual based VMS subgrid modeling for vortex flows. Computer Metods in Applied Mecanics and Engineering, 99:8 89,. [3] S. Brenner and L. R. Scott. Te Matematical Teory of Finite Element Metods. Springer- Verlag, 8. [] W. Cen, M. Gunzburger, D. Sun, and X. Wang. Efficient and long-time accurate second-order metods for Stokes-Darcy system. SIAM Journal of Numerical Analysis, 55:563 58, 3. [5] C. Foias and R. Temam. Gevrey class regularity for te solutions of te Navier-Stokes equations. Journal of Functional Analysis, 87:359 369, 989. [6] T. Gallay and C.E. Wayne. Invariant manifolds and te long-time asymptotics of te Navier-Stokes and vorticity equations on R. Arcive for Rational Mecanics and Analysis, 633:9 58,. [7] K. Galvin, T. Heister, M.A. Olsanskii, and L. Rebolz. Natural vorticity boundary conditions on solid walls. Submitted, 5. [8] T. B Gatski. Review of incompressible fluid flow computations using te vorticity-velocity formulation. Applied Numerical Matematics, 73:7 39, 99. [9] S. Gottlieb, F. Tone, C. Wang, X. Wang, and D. Wirosoetisno. Long time stability of a classical efficient sceme for two-dimensional Navier-Stokes equations. SIAM Journal on Numerical Analysis, 5:6 5,. 8
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