This paper presents a review of results for the recently introduced reduced Navier Stokes-α (rns-α) model of incompressible, viscous flow, given by

Transcription

1 fluids Article Te Reduced NS-α Model for Incompressible Flow: A Review of Recent Progress Abigail L. Bowers and Leo G. Rebolz, * Department of Matematics, Florida Polytecnic University, Lakeland, FL 3385, USA; abowers@flpoly.org Department of Matematical Sciences, Clemson University, Clemson, SC 9634, USA * Correspondence: rebolz@clemson.edu; Tel.: Academic Editor: William Layton Received: 3 June 7; Accepted: 3 June 7; Publised: 6 July 7 Abstract: Tis paper gives a review of recent results for te reduced Navier Stokes-α rns-α model of incompressible flow. Te model was recently developed as a numerical approximation to te well known Navier Stokes-α model, for te purpose of more efficiently computations in te C finite element setting. Its performance in initial numerical tests was remarkable, wic led to analytical studies and furter numerical tests, all of wic provided excellent results. Tis paper reviews te main results establised tus far for rns-α, and presents some open problems for future work. Keywords: incompressible flow; regularization models; α models; turbulence. Introduction Tis paper presents a review of results for te recently introduced reduced Navier Stokes-α rns-α model of incompressible, viscous flow, given by α w t + w t + Dw w + p ν Dw = f, w =, w = w, 3 were w represents velocity, p a Bernoulli-type pressure, f a body force, and D a deconvolution operator intended to approximate te inverse of te Helmoltz also called α differential filter F = α + I. Te parameter α > represents te filtering radius. For simplicity, only van Cittert approximate deconvolution, D := D N = N n= I F n, is considered erein. Tis is te most common type used in fluid flow modeling [ 3]. Oter types of approximate deconvolution operators, suc as Tikonov Lavrentiev, may yield similar results, but all numerical tests done to date for rns-α ave used van Cittert. Some advantages of van Cittert, wic likely make it te most common, are tat it is simple to use, easy to analyze, is formally very ig accuracy, and as performed very well in many computational tests for several models [ 3]. Note tat te groundbreaking idea of using approximate deconvolution in fluid models is due to Stolz, Adams and Kleiser [4 7]. Typically N is cosen small, and for all numerical simulations run so far wit rns-α, N. Fluids 7,, 38; doi:.339/fluids338

2 Fluids 7,, 38 of Te rns-α model was proposed in [8] as a numerical approximation to te well-known NS-α model see e.g., [9 ] tat is more efficiently computable in a C finite element framework. NS-α is given by v t + v v + p ν v = f, 4 v = v =, 5 α v + v v =. 6 Tis model also known as Camassa Holm equations [ 3] as extensive and attractive teoretical properties see [9,] and references terein, including well-posedness [], frame invariance [4], aderence to Kelvin s circulation teorem [9], conserving a model energy and elicity [9], requiring significantly fewer degrees of freedom tan direct numerical simulation DNS of Navier Stokes [9], and accurately predicting scalings of te turbulent boundary layer [5]. Most oter fluid models do not ave all or even most of tese properties, and tus NS-α is widely considered to be a very pysically accurate model. However, ow to construct stable, efficient algoritms for it in te C finite element setting is an open problem. Te essential issue is tat te filtering seemingly cannot be decoupled from te momentum-mass system in a stable way, leaving te practitioner to solve te nonlinear problem at eac time step. Not only does tis create te need for many system solves at eac time step, but if one wises to use a Newton iteration, ten te linear systems must solve simultaneously for te velocity, filtered velocity, and pressure. However, rns-α can be computed efficiently in tis setting, in te sense tat it can stably decouple te filtering from te mass/momentum system, solve only one mass/momentum system at eac time step, and obtain optimal accuracy. Initial testing of rns-α in [8,6,7] revealed outstanding results for turbulent cannel flow simulation and flow past a cylinder. Te purpose of tis paper is to review te recent analytical and numerical results for rns-α. In Section, te derivation of te model is sown and ow it is related to oter models and stabilizations. Section 3 reviews te analytical results establised for te model, including a priori bounds, energy trapping, global well-posedness, energy dissipation rate, and energy spectra. Results for sensitivity of te model wit respect to te filtering radius are discussed in Section 4. Numerical scemes and teir analysis are given in Section 5, bot for rns-α and its associated sensitivity equations from Section 4. Finally, in Section 6, results of numerical tests for cannel flow past a cylinder and turbulent cannel flow are discussed. Section 7 presents some open problems for possible future work.. Derivation of te Model and Connections to Oter Models It will be elpful for te later discussion to present te derivation found in [8] of te rns-α model, wic starts from te NS-α model. Recall from above tat te NS-α model is defined as follows, after writing Fv = v: v t + v Fv + q ν v = f, Fv =. It is discussed in detail in [8] wy tis model is difficult to efficiently compute wit in a C finite element framework. Te rns-α model is created troug a series of transformations to NS-α, as follows. By using I = F F = α + IF in NS-α, one gets α Fv t + Fv t + F Fv Fv + q ν F Fv = f, Fv =.

3 Fluids 7,, 38 3 of Denote w = Fv, wic provides α w t + w t + F w w + q ν F w = f, w =. Next, use te approximation of te deconvolution operator, i.e., tat F w D w, to get te rns-α model, written in terms of w and p after renaming variables due to te approximation:.. Connections to Oter Models α w t + w t + Dw w + p ν Dw = f, w =. Tere are several models closely related to rns-α tat ave been studied recently, including NS-α [9, 3,8,9], NS Voigt [ ], and various oter regularization models. It is important to make connections between te different models wen possible, so existing results of various models can be fully utilized. It is clear from Equations 3 wit D = D N, tat for any N, coosing α = recovers te Navier Stokes equations. Tus, if a mes sufficient for a DNS of Navier Stokes is used wit rns-α and α is cosen on te order of te mes widt, ten te solution is expected to matc te Navier Stokes solution closely. Deconvolution teory from [] sows tat N = formally recovers NS-α, and te coice N = recovers te NS Voigt model [ ]. Hence, one can interpret te rns-α model as being in between NS-α and NS Voigt, in te sense of deconvolution order. Tese connections are summarized in te following Table. Table. Connections summary. α N rns-α Model Reduces to [, ] Navier Stokes > NS Voigt > NS-α A second important connection for rns-α is to te generalized α-models studied in [3]. By replacing w wit Fv and using te notation of [3], rns-α fits te general form wit A = ν DF, M = F, N = DF, and χ =. Work done for tese models reveals teir strengts and weaknesses, and tus identifying connections between models is critical, as tey can be extended to rns-α. For example, te main arguments in [8] for well-posedness of rns-α are similar to tose made for NS Voigt in [,4]. Wen compared to oter α-models, rns-α as displayed better results on wall bounded turbulence, as demonstrated in [8,6,5]. A key difference between tese models lies in te viscous term, wic as te form ν D N w instead of te standard ν w. At te continuous level, te rns-α viscosity provides no extra smooting, but it does provide additional dispersion wen compared to te standard viscous term. Tis is due to te fact tat D N, and tis can be seen at ig wave numbers since ere D N N + []. Furter insigt is gained by expanding te deconvolution operator of te viscous term. Following [6], one can write ν D N w = ν w ν D N w w 7 = ν w ν D N w 8 = ν w + α ν D N w, 9

4 Fluids 7,, 38 4 of were w := w w can be considered as fluctuations about te filtered mean velocity []. In Equation 8, te extra term ν D N w reveals a connection to variational multiscale models VMMs. It takes te form of a viscosity for velocity fluctuations, wic is in te same spirit as VMMs, but differs in tat ere fluctuations are defined wit filtered quantities as means. Interestingly, similar dissipation/penalty/stabilizations for te advection equation [7] and in turbulence simulations [8] were recently considered. Tese were also constructed by using filtering to define means and fluctuations, and ave proven to be very successful at increasing accuracy of simulations on coarse discretizations. One can also observe te effect of te extra term on te energy balance. Assuming periodic or no-slip boundary conditions for bot te velocity and filtered velocities and using Equation 9, test rns-α wit w to obtain d α w + w + ν w + να D / dt N w, D/ N w = f, w, and using te filter definition yields te energy equality from regularity results in [8], all operations are justified d α w + w + ν w + να D / dt N w + να 4 D / N w = f, w. Tis demonstrates te effect on te energy dissipation, wic serves to dissipate iger order norms of D / N w. Applying te filter definition once again, one can write d α w + w + ν w + να D / dt N w w + να 4 D / N w w = f, w, wic more clearly sows tat tis extra viscous term acts to dissipate velocity fluctuations. 3. Analytical Results Te first analysis of rns-α was performed in [8], were it was sown tat te model is well-posed for a fixed end time, and tat regularity of solutions depended on te regularity of te data. Global in time energy and regularity results were proven in [7], and are stated below, after some initial preliminaries. Te treatment of energy by te model was studied in [6]. Results for te energy conservation of te model, energy spectra, and energy dissipation were all very good: an appropriate energy quantity is conserved, te model exibits a k 5/3 energy cascade on te large scales in te inertial range and at k 3 on smaller scales. Furtermore, energy is dissipated at te rate U3 L, independent of te Reynolds number, wic is consistent wit true fluid flow. Consider in tis section a domain Ω R d, d = or 3, wic is a box. Te notation and, denotes te L Ω norm and inner product, wit all oter norms being labeled clearly. Assume periodic boundary conditions on te box, wic is te common setting for suc energy studies, as it is typically te only case were well-posedness of solutions is known. For te energy balance and dissipation results, te results extend to te wall-bounded case if te well-posedness results also extend to te wall-bounded case, wic is an open problem. Te energy spectra results, owever, rely eavily on te periodic setting, as tey explicitly use Fourier decompositions. Most of te filtering and deconvolution results can be extended to te case of omogeneous Diriclet boundary conditions, if filtered velocities satisfy no-slip boundary conditions. Tis is not widely accepted as a correct boundary condition for te filter on te oter and, tis is te boundary condition use in all successful turbulent flow simulations wit rns-α [8,6], and seems to work quite well. Denote X, Q := H # Ω, L # Ω H Ω, L Ω te periodic, zero-mean velocity and pressure spaces. Te space H Ω is te dual space of X.

5 Fluids 7,, 38 5 of 3.. Energy Bounds and Well-Posedness Presented now are several long time energy bounds. Te following was proven in [7], and sows te energy is trapped for all times if f L, ; H Ω and w X. Lemma Energy trapping. Suppose f L, ; H Ω and w X. Ten, t >, { } α wt + wt max ν β f L,,H, α w + w := C E, { } were β = min, C α Ω. Tus, te kinetic energy is contained in te ball B, C E for all time. Under te additional restriction tat f L, ; H Ω, ten energy must decay to. Tis is proven in [7], and stated in te following lemma. Lemma Energy decay. Suppose f L, ; H Ω and w X. Ten, te energy decays to zero as t : lim α wt + wt =. t Global well-posedness of te model was able to be proven, using te long time a priori energy bounds above [7]. Lemma 3 Existence and uniqueness of weak solutions. Suppose f L, ; H Ω or f L, ; H Ω and w X. Ten, tere exists a global unique weak solution to Equations 3. Additional regularity of te data leads to global in time iger order regularity of solutions, and is proven in [7]. Lemma 4 Higer order estimates. Assume a solenoidal initial condition w H Ω X. If f L,, L Ω, ten w L,, H Ω. If f L,, L Ω, ten w H as t. Remark. Assuming iger order regularity of te data, iger order regularity of solutions can be establised using bootstrapping and te tecniques used in [7]. 3.. Energy and Helicity Balances Integral invariants suc as energy and elicity are fundamental for a priori estimates used in existence teorems. Tey can also provide pysical insigt into a model s beavior, as well as its pysical relevance. It as long been establised tat tese quantities are invariant in true fluid flow i.e., in te Navier Stokes equations, and as tey are believed to be fundamental to te organization of flow structures, a good model sould capture tese quantities accurately. Te conserved model energy and elicity take te form: E rnsα t := Ω H rnsα t := Ω α wt + wt, α D / N wt, D/ N wt + D/ N wt, D/ N wt. Te energy of tis model is te same as for NS Voigt []. As w represents an approximation of v, wit v being te NS-α velocity, one can observe tat te rns-α energy is analogous to te NS-α energy E NSα := Ω α vt + vt [9,]. By similar reasoning, te rns-α elicity is connected to te NS Voigt elicity and NS-α elicity.

6 Fluids 7,, 38 6 of It is stated below, and proven in [6], tat te energy and elicity of te model are preserved. Furtermore, if te forcing is only spatially dependent, ten te energy balance leads to a global in time bound on energy. Teorem. Let w be te solution to Equations 3 under periodic boundary conditions, wit sufficiently smoot data. Ten, te following model energy and elicity balances old: E rnsα T + T Ω ν D / N wt dt = E rnsα + T f t, wt dt, Ω HT + T Ω ν D N wt, D N wt dt = H + T f t, D Ω N wt dt. In te case wit vanising viscosity and no external forcing, te model energy and elicity are exactly conserved. Tat is, for v = and f =, E rnsα t = E rnsα, H rnsα t = H rnsα. If te forcing is only spatially dependent, te energy balance from Teorem can be furter analyzed, revealing tat ɛ rnsα := ν Ω D/ N w is te energy dissipation of te model. Te following result is proven in [6]. Suppose tat te forcing is solenoidal and only dependent on space, f x, t = f x L Ω, and te initial condition w H Ω. Ten, T T ɛ rnsαt dt = T T 3.3. Energy Dissipation Rate sup E rnsα t Cdata <, 3 t, ν Ω D/ N w dt T E rnsα + f T T Ω / Ω w /. 4 Consider now te rate of energy dissipation by te model, and assume a constant solenoidal forcing, i.e., f = f x H Ω, and f =. Te scale of te body force, large scale velocity, and lengt are defined as: F := / Ω f x, U := Ω wx, t /, L := min F, f F /, Ω f T were φ := lim sup T T φt dt denotes te time average. It can be sown tat eac component of te definition for L as units of lengt. Te following teorem for te scaling of te time averaged energy dissipation is proven in [6]. Recall tat N is assumed small, i.e., N 5. Teorem. Te time averaged energy dissipation of te rns-α model is bounded as ɛ rnsα = ν Ω D/ N w C N were C N is a constant dependent only on N e.g., C N 6 wen N =. 3 + Re U 3 L,

7 Fluids 7,, 38 7 of Remark. Te above result is consistent wit K4 penomenology, wic predicts time averaged energy dissipation to scale wit U3 L [9], wit te scalings derived from te Navier Stokes equations directly [3 3], and for te usual NS-α model [33] Energy Spectra It is a generally accepted teory in turbulence tat energy is input at large scales, cascaded but preserved by te nonlinearity troug te inertial range of intermediate scales, and ten dissipated exponentially fast by viscosity in te small scale range. For accurate computations, it is critical to resolve te flow to were viscosity takes over. For te Navier Stokes equations, resolving all of tese scales is a computationally intractable problem, and it is necessary to model. Studying te energy spectra of a model allows us to gain insigt into its computability, as a successful model must ave an inertial range tat is sorter tan tat of te Navier Stokes. Te analysis and notation below follows studies done in [9,34 36]. Decomposing velocity into its Fourier modes yields te balance of energy d dt w k, w k + α w k, w k + ν D N kw k, w k = T k T k, 5 were D N k is te Fourier transform of D N, and wit T k = D N kw < k w k, w k + D N kw k + w > k w k + w > k, w< k, w < k = w j, w > k j <k = w j. j k Te quantity T k T k represents net energy transferred into te wave numbers between [k, k. Time averaging te balance Equation 5 gives te energy transfer equation ν D N kw k, w k = T k T k. 6 Define te model energy of a size k eddy, based on te rns-α model energy E rnsα, as and combining wit Equation 6 gives E rnsα k = + α k ŵ j, j =k T k T k νk3 D N ke rnsα k + α k νk3 N + E rnsα k + α k, wit te last relation coming from []: D N k N + for sufficiently large k. Tis is a key estimate for determining te inertial range, since ere one expects no leakage of energy troug dissipation, and ence T k T k. Tis suggests tat increasing N leads to a sorter inertial range, wic is consistent wit te energy spectrum computations in Section 4. Te inertial range s kinetic energy distribution is now considered. Begin by defining te average velocity of a size k eddy, and ten relate it to model energy: k U k = w k, w k / k / E rnsα k + α k kernsα k / + α k.

8 Fluids 7,, 38 8 of From te Kraicnan energy cascade teory [37], one obtains tat te eddy turnover time is τ k := ku k = + k α / k 3/ E rnsα k /, and tus te model energy dissipation rate scales like ɛ rnsα τ k k Solving for E rnsα k, one obtains te relation k E rnsα k k5/ E rnsα k 3/ + k α /. E rnsα k ɛ/3 rnsα + α k /3 k 5/3, and tis yields a spectrum for kinetic energy E = w : Ek E rnsαk + α k ɛ /3 rnsα k 5/3 + α k /3. Split te spectrum into two parts: if kα > O, ten but if kα < O, Ek ɛ /3 rnsα k 5/3 α k Ek ɛ/3 rnsα k 5/3. ɛ/3 rnsα /3 k 3 α 4/3, Tese scalings sow tat on larger inertial range scales, a k 5/3 rolloff of energy is expected wic agrees wit true fluid flow. However, for wave numbers bigger tan Oα, a rolloff of k 3 is expected. Tese rolloffs are identical to usual NS-α on bot te larger and smaller scales in te inertial range [9]. Tis implies tat significantly less energy is eld in iger wave numbers compared to a Navier Stokes DNS, wic makes te model more computable tan te Navier-Stokes Equations NSE. Numerical tests in [6] illustrate tese scalings. Anoter important takeaway is tat since te rolloffs mirror tose of NS-α, one sould expect computational cost in terms of mes widt resolution to be very similar NS-α, at least in tis periodic setting. Tis means tat total cost in te C finite element setting sould be muc less for rns-α compared to NS-α, since te former is more efficiently computable, as discussed above. 4. Sensitivity of te Model wit Respect to te Filtering Radius α Te parameter α in rns-α represents a filtering radius, and canging α in computations can lead to very different solutions. Error analysis of te model s discretization above suggests α = O is a good coice, and coosing α muc smaller tan tis would lead to no regularization, since suc a discrete filter would effectively ave filtering radius. Trougout te literature, α is often cosen sligtly larger tan, suc as or 6 our experience makes us favor te coice of. Due to tese various coices, it makes sense to study sensitivity of te model to canges in α, and in tis section, a sensitivity equation for α w is derived and analyzed. In later sections, efficient algoritms are proposed for computing it, and finally computations are performed. Tis section follows work done in [7]. Te sensitivity equation for α w is derived by applying α to rns-α, ten setting s = w/ α and r = p/ α. Tis gives

9 Fluids 7,, 38 9 of α s t + s t + α D Nw w + α D Nw s + r ν α D Nw = α w t, 7 s =, 8 s =. 9 Tis system is reduced by eliminating variables for filtered velocities sensitivities after evaluating α D Nw, and writing te system in terms of s = w/ α. Te system depends on N, and for te first few values of N, one gets Dw N D N w α I Ds w w Ds α w 3w 3 w + w Ds 6α w + 4α w 3 4w 6 w + 4 w 3 w Ds α w + 6α w 8α w Note tat for general N, one can write D Nw α can be written as D N s plus a linear combination of filtered w s, N D N w = D N s + α α β N i F i+ w, were te β N i s are integers depending on N. Tus, te general system takes te form α s t + s t + D N s w + D N w s + r ν D N s = α w t + να N i= i= β N i F i+ w α N i= β N i F i+ w w, s =, s =. Tis system is proven in [7] to be well-posed in te periodic setting, and te result reads as follows: Teorem 3 Existence and uniqueness of weak sensitivity solutions. Let f L, ; L Ω and w X. Ten, weak solutions to Equations exist uniquely, and satisfy for T <, T α st + st + ν st dt Cdata. In te numerical tests section below, efficient numerical scemes for computing tis sensitivity system are proposed and analyzed. 5. Numerical Scemes and Analysis An efficient numerical sceme for approximating solutions to rns-α using a C finite element spatial discretization, togeter wit an implicit-explicit BDF timestepping, is given. Tis time discretization linearizes te system at eac time step and decouples te filtering/deconvolution equations from te momentum/mass system. After providing some preliminaries, te sceme and results for its stability, well-posedness, and convergence, are presented. 5.. Problem Setting and Preliminaries Assume a regular, conforming triangulation/tetraedralization τ Ω, and associated inf-sup stable discrete velocity-pressure spaces X, Q X, Q tat are piecewise polynomials on eac element. Define te discretely divergence free subspace by

10 Fluids 7,, 38 of V := {v X, v, q = q Q }. Discrete filtering is defined by te standard finite element discretization of te α filter: Given φ L Ω, find F φ = φ X satisfying α φ, χ + φ, χ = φ, χ χ X. Discrete van Cittert deconvolution is analogous to te continuous case, but defined wit te discrete filter: D N = N n=i F n. An explicit filter operator is now given tat filters known quantities exactly, but for unknown quantities, it instead filters a known second order approximation: F w j = F w j for j=,,...,n, and F w n+ Tis operator defines an explicit deconvolution operator D N wn+ = N n= I F n. := F w n wn. Finally, te implicit-explicit IMEX BDF, C finite element sceme for te rns-α model can be stated. Algoritm : [BDF] Given endtime T >, timestep t >, forcing f L, T; H Ω, and initial velocities w, w V, set M = t T and for n =,,..., M, find wn+, q n+ X, Q satisfying for all v, r X, Q, α 3w n+ t 4w n + wn, v + 3w n+ t 4w n + wn, v + DN wn wn w n+, v q n+, v + ν D N wn+, v = f t n+, v, 3 w n+, r = Analysis of te Sceme Results of te stability and convergence of Algoritm are now stated, and come from [8]. Lemma 5 Stability. Suppose tat t < α w M + w M M + n= α. Ten solutions of Algoritm satisfy 4CNν w n+ w n + wn + ν t M n= w n Cdata. 5 Moreover, solutions to Algoritm exist uniquely since te sceme is linear at eac timestep. Tus, te algoritm is well-posed.

11 Fluids 7,, 38 of Remark 3. In most settings were large eddy simulation LES models suc as rns-α is used, it generally olds tat ν α. Moreover, typically N 3, giving CN 6. Tus, in practice, te timestep restriction will typically be mild. Convergence of te sceme is proven in [8] and is stated in te following teorem. Teorem 4. Suppose X, Q are cosen to be eiter P k, P k Taylor Hood elements or P k, P disc k Scott Vogelius elements and suppose furter tat w, p is a solution of Equations and for a given α >, ν >, wit forcing f L, T; H Ω and initial velocity w V, satisfying regularity criteria w L, T; H k+ Ω H 3 Ω, p L, T; H k Ω, w t, w tt, w ttt L, T; H k+ Ω H 3 Ω. Suppose furter tat te stability criterion for te timestep size is satisfied. Ten te error of Algoritm in approximating solutions to te rns-α model satisfies wt w M + M n= +α wt w M + ν t 5.3. Numerical Sceme for rns-α Sensitivity wt n+ wt n + wt n w n+ w n + wn M n= / wt n w n C t + k. 6 An efficient numerical sceme is now detailed for te rns-α sensitivity system to be used togeter wit te IMEX BDF sceme above. Te key feature for efficiency is tat te discrete velocity/pressure and sensitivity systems will ave exactly te same coefficient matrices. Tus, preconditioners for te velocity solve can be reused, and also one can take advantage of block solvers tat work wit multiple rigt-and sides, since te proposed algoritms allows solving for s n and w n+ simultaneously. Te discrete sceme is now presented. Step is exactly te rns-α sceme from above. Step solves for velocity and pressure sensitivity, and is decoupled from Step. Algoritm : [rns-α wit sensitivity] Given w, w V, s = s = V, f L, T; L Ω, and timestep t = M T, for n =, 3,..., M, Step : Find w n+ V satisfying: 3w n+ t 4w n + wn, v + α t + DN wn wn w n+, v + ν 3w n+ 4w n + wn, v D N wn+, v = f t n+, v 7 Step : Find s n+ V satisfying: 3s n+ t 4s n + sn = α t α + α 3s n+ t 4s n + sn, v + DN wn wn s n+ 3w n+ N i= β N i F i+ w n+ w n+, v, v + ν να, v D N sn+, v 4w n + wn, v DN sn sn w n+, v N i= β N i F i+ w n+, v 8

12 Fluids 7,, 38 of Stability and convergence of te Step is stated above, and it is proven in [7] tat Step is stable and convergent. However, tere is a sligtly more restrictive condition on te time step size. Teorem 5. Let w H Ω, f L [, T], L Ω, and coose t suc tat t < min{ α 4CNν,O/3 }. Ten, s M + α s M M + ν t s n+ n= Cdata. Furtermore, assuming sufficiently smoot data, te convergence estimate olds: 6. Numerical Results st s M + ν t M n= st n s n C t + k. Results of computations for rns-α are presented in tis section. First, results of convergence rate tests are given, to illustrate te convergence results above. Next, two application problems are considered, D flow past a cylinder and 3D turbulent cannel flow. For te two application problems, comparisons of te model solutions to te literature are made, and sensitivities are also computed and discussed. 6.. Verification of Convergence Rates In [8], convergence rate verification tests were performed for te IMEX sceme above for rns-α. Te test problem used was an exact rns-α solution i.e., it solves and wit f = given by w = cosnπx sinnπy sinnπx cosnπy C DN n π νt e +n π α, 9 were C D is defined for eac N by D N w = C DN w justification of te existence of suc a constant is given in [8]. Approximations to tis exact solution were calculated in [8] on Ω =, and t T =., using Algoritm, wit f =, N =, n =, ν =, and α = 6 wit C D =. Errors and +π α rates for te sceme were computed, using successively refined uniform meses and timesteps, wit P, P Taylor Hood elements. Based on Teorem 4, convergence is expected to satisfy w w, := w w L,T;H Ω = O + t. Tus, a convergence rate of is predicted if te mes widt and te time step are reduced togeter, by a factor of at eac refinement, wic is observed to be te case from Table. Table. Velocity errors and rates for te convergence rate test are given in te table, and are consistent wit te optimal teoretical rate of. t w w, Rate /4 /.5 - /8 / /6 / /3 / /64 /

13 Fluids 7,, 38 3 of 6.. D Cannel Flow Past a Cylinder Results of tests of te rns-α model/sceme for D flow past a cylinder are now presented. Tis is a bencmark problem from [38 4], wit te domain being a..4 rectangular cannel wit a radius =.5 cylinder, centered at.,., as sown in Figure. No slip boundary conditions are prescribed on te walls and cylinder, and te inflow and outflow profiles are given by u, y, t = u., y, t = 6 sinπ t/8y.4 y,.4 u, y, t = u., y, t =. Te kinematic viscosity is set as ν = 3, te forcing f =, and te flow starts from rest. Te correct beavior is for a vortex street to form by t = 4 and continue to t = 8. Te flow is driven by an interaction of te fluid wit te cylinder, wic is an important scenario for industrial flows. Tis flow is not turbulent, but is still very callenging for most numerical models and metods, especially on coarser meses Figure. Sown above is te cannel flow around a cylinder domain. Important statistics for tis flow are te predicted lift and drag forces. For a velocity field u and pressure p, lift and drag coefficients are defined by c d t = c l t = ρlumax S ρlu max S ρν u t S t n y ptn x ds, n ρν u t S t n x ptn y ds, n were u ts is tangential velocity, U max = is te max velocity at te inlet, L =., ρ =, S is te cylinder, and n = n x, n y is te outward unit normal. Solutions were computed using Algoritm wit t =. and P, P disc elements on a very coarse barycenter mes tat provided 54 velocity degrees of freedom dof. Tis coice of elements is made because te model is implemented using a Bernoulli-type pressure, wic is significantly more complex tan usual pressure and is known to affect te overall error if standard element coices are used [4], but not if divergence-free elements are used. Te deconvolution parameter N = was cosen, and tests were run wit α varying from. to.6. To evaluate tese solutions, values for te maximum drag c d,max and lift c l,max coefficients were calculated, and compared to tose found in te resolved bencmark tests of [4]: c re f d,max = , cre f l,max =

14 Fluids 7,, 38 4 of For comparison, te max lift and drag coefficient values were calculated on te same mes, elements, and time step by te NSE discretized by te following extrapolated Crank Nicolson finite element metod: u n+ t u n, v + 3 un un p n+/, v + ν u n+/, v u n+/, v = f t n+/, v, 3 u n+, q =. 3 One observes from Table 3 tat rns-α provides muc better max lift and drag coefficients tan te coarse mes NSE. Te reference values come from upwards of 7 dof simulations, and it is remarkable te rns-α gets tis level of accuracy on suc a coarse discretization. Table 3. Sown in te table are max lift and drag coefficients for various models and parameters, for D flow past a cylinder. Model α c d,max % Error c l,max % Error Navier Stokes NS % % rns-α %.5 6.9% rns-α % % rns-α %.5 4.8% rns-α % % rns-α %.49 3.% rns-α %.487.9% 6... Sensitivity to α in Flow Past a Cylinder For tis test problem, sensitivity of te solution to α was considered, and Algoritm was computed wit α =., t =., and P, P disc Scott Vogelius elements on a barycenter refined Delaunay triangulation, wic provided 35,948 total dof. Plots of te sensitivity magnitude are sown in Figure at various times. At early times, te sensitivity is created at te cylinder, and ten as time progresses, it gets convected into te flow. By t = 6, values of s are near 6. Creation of sensitivity at te solid boundary is not unexpected, since te sensitivity equation resembles a vorticity equation. t =.4 t =.8 t =. t = 4 t = 6 Figure. Sensitivity magnitudes at various times.

15 Fluids 7,, 38 5 of Lift and drag sensitivities are also calculated, defined by differentiating te lift and drag formulas above wit respect to α. For te tests above, tese quantities are also calculated: s d t = s l t = ρlumax S ρlu max S ρν s t S t n n y rtn x ds, ρν s t S t n n x rtn y ds. Here, s ts denotes te tangential sensitivity. Sown in Figure 3 are plots of s d and s l versus time tat came from tis test. One observes very large sensitivity wit te lift, and less so in te drag, but still significant. One also observes tat te drag sensitivity is positive for most of te simulation, indicating tat increasing α will increase drag. Lift sensitivity s l t Drag sensitivity s d t Figure 3. Sensitivities of lift and drag coefficients, plotted wit time Turbulent Cannel Flow at Re τ = 59 Results from rns-α tests on a bencmark turbulent cannel flow problem wit friction Reynolds number Re τ = 59 from [6] are now discussed. Te results below sow tat rns-α does an outstanding job at matcing te DNS mean velocity profile on a very coarse mes. To our knowledge, no oter model produces results tis good on suc a coarse mes. For te widely cited DNS results [43], te number of degrees of freedom used was 3 million our tests use just 8,8. For comparison, results on te same discretization for Leray-α, NS-α, and NS Voigt models were computed on te same coarse mes in [6] and sown to be muc worse tan rns-α Problem Description and Results Te problem setup is given in detail in [8,6,44], and te interested reader is referred to tose works for additional details. Te domain is te box Ω = π, π, π/3, π/3, wit omogeneous Diriclet boundary conditions enforced on te top and bottom walls y = and y =, and periodic boundary conditions enforced on te remaining sides. Te kinematic viscosity is set at ν = Re τ. Te initial condition is created by randomly perturbing te DNS average velocity profile pointwise. Tese perturbations create te turbulence as te simulation is run. Te forcing is defined by f =,, T, and is dynamically adjusted at eac time step to maintain te bulk velocity. Te simulations are run to t = 6, and te time and space average of te velocity is done from t = to t = 6. P 3, P disc divergence-free Scott Vogelius elements are used wit a barycentric tetraedral mes tat is weigted towards te walls, and provides 8,8 velocity dof. Te time step size t =., deconvolution N =, α was varied. Te value of α =. gives an optimal to witin.5 solution in te L sense of best matcing te DNS mean velocity profile; note tat tis value is close to te mes resolution near te wall. Mean streamwise velocity profiles for rns-α are sown in Figure 4, togeter wit DNS data. Results tis good are seemingly unique for any regularization model on suc a coarse mes, using a finite element discretization.

16 Fluids 7,, 38 6 of Te L, difference between te DNS mean streamwise velocity, and various model s mean streamwise velocity, is sown in Table 4 te values of α are optimal to witin.5 in te interval [,.] One immediately observes te rns-α is muc more accurate tan Leray-α and NS Voigt wic bot barely improve on te no model, i.e., NSE on te coarse mes. NS-α blows up for eac coice of α note te nonlinear problem was not solved at eac time step; instead, a linearization on te filtered term in te nonlinearity is done, to make its efficiency te same as te oter models see [6]. 5 DNS rns 5 DNS rns U mean 5 5 U mean y a 4 3 y b Figure 4. Sown above are te average velocity profiles of te coarse mes reduced Navier Stokes-α rns-α simulation and direct numerical simulation DNS of te Navier-Stokes Equations NSE, at Re τ = 59, sown in regular coordinates in a, and near te wall in a log scale in b. Table 4. Te L, difference between te model s predicted velocity profile and te cubic spline of Moser, Kim, and Moin velocity profile for Re τ = 59, using results from te model wit optimal α from. to.7, wit increments of.. No NS-α simulation wit te implicit-explicit IMEX algoritm from te appendix remained stable for te entire simulation. Re τ = 59 Model Optimal α u DNS y u model y L, Navier-Stokes Equations NSE coarse mes -.49 rns-α..84 NS-α DNE DNE Leray-α.4.3 NS Voigt Model Sensitivity for Turbulent Cannel Flow wit Re τ = 59 rns-α sensitivity was tested in [7] for tis bencmark turbulent cannel flow problem. Since te quantity of interest for tis problem is te long time averaged velocity, a sensitivity study of rns-α for tis problem is now performed, to determine te long time averaged velocity sensitivity. Te same discretization as in [6] is used for tis test: decoupled IMEX-BDF-finite element discretization step of Algoritm, and P 3, P disc Scott Vogelius elements on a tetraedral mes tat provided 8,8 velocity and sensitivity degrees of freedom. Bot N = and N = wic is te NS Voigt model are tested, and α =.,.5 and.. Starting from te T = 6, N =, α =.5 turbulent solution as te initial condition created using te setup above, te simulation is run to T = 7 s using a time step size of t =. total time steps. Algoritm step is used to calculate sensitivities at eac time step, and time and space x and z directions averages are taken to get te mean streamwise velocity and sensitivity profiles. From Figure 5, observe tat rns-α N = gives an excellent prediction of te mean streamwise velocity profile for eac coice of α, significantly better tan te NS Voigt N = case. For te

17 Fluids 7,, 38 7 of varying α, tere is no visible cange in te NS Voigt solution, and a sligt visible cange in te rns-α solution. Figure 6 sows te mean streamwise sensitivity and relative sensitivity magnitudes, for bot rns-α and NS Voigt. One observes tat rns-α is more sensitive tan NS Voigt to canges in α. More interesting is te observation tat te sensitivity is largest near te boundary. Altoug tis is peraps expected due to te boundary layer is critical for turbulent cannel flow; and te correct boundary conditions for filtered velocities is an open problem for LES/α models note te N = model does not use filtered velocity boundary conditions..5 DNS Voigt =.5 rns =.5.5 DNS Voigt =. Voigt =.5 Voigt =..5 DNS rns =. rns =.5 rns =. y y y <U> <U> <U> Figure 5. Average streamwise velocity profiles for DNS, rns-α, and NS Voigt models..5 Voigt =.5 rns =.5.5 Voigt =.5 rns =.5 y y <S> 4 6 <S>/<U> a b Figure 6. Average streamwise sensitivity a and relative streamwise sensitivity magnitude b for rns-α and Voigt solutions wit α = Conclusions Te main results to date for rns-α ave been reviewed in tis paper. Tis includes global well-posedness, energy spectrum analysis, and energy dissipation analysis. An efficient numerical discretization is reviewed, and corresponding stability and convergence results are given. Additionally, sensitivity equations for rns-α are derived, and a numerical sceme for tem is given. Finally, results of several numerical tests are given, wic sow excellent results on several bencmark problems. Tere are several directions for future work. Furter testing, furter comparison to oter LES models, and exploration of te model outside of te finite element framework sould all be done. One could also immediately move tis model into te multipysics arena, for non-isotermal flow, or doubly diffusive flow, or even to magnetoydrodynamics. Additional numerical testing on difficult bencmark problems is also important for any newer models. An interesting study could likely be

18 Fluids 7,, 38 8 of done for rns-α s near-wall beavior; it remains unclear wat makes rns-α so muc more accurate for turbulent cannel flow, but one ypotesis is tat it could be more accurate near te wall. Anoter interesting future direction of tis work is to consider te rns-α model, but remove all modeling except in te viscous term. Tat is, one could consider a simplified model to be w t + w w + p ν D N w = f, 3 w =, 33 or possibly keep te Voigt term α w t. Proceeding as in Section, one can write te momentum equation of tis reduced model as w t + w w + p ν w ν D N w w = f. Note tat tis system can be considered as Navier Stokes wit an additional term meant to dissipate velocity fluctuations. Our discussion in Section sows tat te altered viscous term is strongly related to te VMM approac to turbulence modeling, so it is possible tat it is only tis term tat is to tank for te model s successes on te turbulent cannel flow bencmark tests. One also notes tat implementing te term would require very little extra work in a simulation, if te velocity term is treated implicitly and te filtered velocity is treated explicitly by extrapolating from previous time steps. Anoter possible direction for future work is coosing α locally and dynamically. Work in [45 47] as sown tat carefully cosen local filtering radii can give large increases in accuracy in certain settings, and so extending tese ideas to rns-α seems like a logical direction. Autor Contributions: Abigail L. Bowers and Leo G. Rebolz did an equal amount of work on tis manuscript, including te analysis, experiments, and writeup. Conflicts of Interest: Te autors declare no conflict of interest. References. Berselli, L.; Iliescu, T.; Layton, W. Matematics of Large Eddy Simulation of Turbulent Flows; Scientific Computation; Springer: Berlin, Germany, 6.. Cacon-Rebello, T.; Lewandowski, R. Matematical and Numerical Foundations of Turbulence Models and Applications; Springer: New York, NY, USA, Layton, W.; Rebolz, L. Approximate Deconvolution Models of Turbulence: Analysis, Penomenology and Numerical Analysis; Springer: Berlin, Germany,. 4. Stolz, S.; Adams, N.; Kleiser, L. Te approximate deconvolution model for large-eddy simulations of compressible flows and its application to sock-turbulent-boundary-layer interaction. Pys. Fluids, doi:.63/ Adams, N.; Stolz, S. A subgrid-scale deconvolution approac for sock capturing. J. Comput. Pys., 78, Adams, N.A.; Stolz, S. On te approximate deconvolution procedure for LES. Pys. Fluids 999,, Stolz, S.; Adams, N.; Kleiser, L. An approximate deconvolution model for large-eddy simulations wit application to incompressible wall-bounded flows. Pys. Fluids, 3, Cuff, V.; Dunca, A.; Manica, C.; Rebolz, L. Te reduced order NS-α model for incompressible flow: Teory, numerical analysis and bencmark testing. ESAIM Mat. Model. Numer. Anal. 5, 49, Foias, C.; Holm, D.; Titi, E. Te Navier Stokes-alpa model of fluid turbulence. Pysica D, 5, Foias, C.; Holm, D.; Titi, E. Te tree dimensional viscous Camassa Holm equations, and teir relation to te Navier Stokes equations and turbulence teory. J. Dyn. Differ. Equ., 4, 35.. Cen, S.; Foias, C.; Holm, D.; Olson, E.; Titi, E.; Wynne, S. Te Camassa Holm equations as a closure model for turbulent cannel and pipe flow. Pys. Rev. Lett. 998, 8, Cen, S.; Foias, C.; Olson, E.; Titi, E.; Wynne, W. Te Camassa-Holm equations and turbulence. Pysica D 999, 33,

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20 Fluids 7,, 38 of 4. Jon, V.; Linke, A.; Merdon, C.; Neilan, M.; Rebolz, L.G. On te divergence constraint in mixed finite element metods for incompressible flows. SIAM Rev. 7, in press. 4. Jon, V.; Rang, J. Adaptive time step control for te incompressible Navier Stokes equations. Comput. Metods Appl. Mec. Eng., 99, Moser, R.; Kim, J.; Mansour, N. Direct numerical simulation of turbulent cannel flow up to Re τ = 59. Pys. Fluids 999,, Jon, V.; Roland, M. Simulations of te turbulent cannel flow at Re τ = 8 wit projection-based finite element variational multiscale metods. Int. J. Numer. Metods Fluids 7, 55, Galvin, K.; Rebolz, L.; Trencea, C. Efficient, unconditionally stable, and optimally accurate FE algoritms for approximate deconvolution models. SIAM J. Numer. Anal. 4, 5, Bowers, A.; Rebolz, L. Numerical study of a regularization model for incompressible flow wit deconvolution-based adaptive nonlinear filtering. Comput. Metods Appl. Mec. Eng. 3, 58,. 47. Layton, W.; Rebolz, L.; Trencea, C. Modular nonlinear filter stabilization of metods for iger Reynolds numbers flow. J. Mat. Fluid. Mec., 4, c 7 by te autors. Licensee MDPI, Basel, Switzerland. Tis article is an open access article distributed under te terms and conditions of te Creative Commons Attribution CC BY license ttp://creativecommons.org/licenses/by/4./.