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Computability and adaptivity in CFD

J. Hoffman1 and C. Johnson2

1 School of Computer Science and Communication, KTH, SE-10044 Stockholm, Sweden2 Department of Applied Mechanics, Chalmers University, SE-41296 Goteborg, Sweden

Abstract

We briefly recall research on adaptive computational methods for laminar compressible and incompressible flow,and we then move on to present recent work on computability and adaptivity for turbulent incompressible flow,based on adaptive stabilized Galerkin finite element methods with duality based a posteriori error control forchosen output quantities of interest, here referred to as General Galerkin G2 methods. We show in concreteexamples that outputs such as mean values in time of drag and lift of turbulent flow around a bluff body arecomputable on a PC with a tolerance of a few percent using a few hundred thousand mesh points in space.

1. Introduction

The Navier-Stokes (NS) equations form the basic mathematical model in fluid mechanics and describea large variety of phenomena of fluid flow occurring in hydro- and aero-dynamics, processing industry,biology, oceanography, meteorology, geophysics and astrophysics. Fluid flow may contain featuresof incompressible and compressible flow, Newtonian and non-Newtonian flow, and turbulent andlaminar flow, with turbulent flow being irregular with rapid fluctuations in space and time and laminarflow being more organized. Computational Fluid Dynamics CFD concerns the digital/computationalsimulation of fluid flow by solving the Navier-Stokes equations numerically. In these notes we start byrecapturing work on adaptive methods for laminar flow, and we then focus on recent developments onadaptive methods for turbulent incompressible Newtonian flow.

The basic problems of CFD is computability relating to errors from numerical computation,and predictability relating to errors from imprecision in given data. The basic question ofcomputability/predictability for a given flow may be formulated as follows: what quantity can becomputed/predicted to what tolerance to what cost? We emphasize the quantitative aspects concerningboth the choice of quantity of interest, or output, the error tolerance and the cost. For computabilitythe cost reflects the precision of the computation with direct connection to the computational work(number of arithmetical operations and memory requirements), and for predictability the cost reflectsthe required precision of data. We expect a turbulent flow to be more costly than a laminar flow, anda pointwise quantity (e.g the viscous stresses at specific points) to be more costly than an averagequantity (e.g. the drag or lift), and of course we expect the cost to increase with decreasing tolerance.

The Reynolds number Re = ULν , where U is a characteristic flow velocity, L a characteristic length

scale, and ν the viscosity of the fluid, is used to characterize different flow regimes. If Re is relativelysmall (Re ≤ 10 − 100), then the flow is viscous and the flow field is ordered and smooth or laminar,while for largerRe, the flow will at least partly be turbulent with time-dependent non-ordered features

Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Rene de Borst and Thomas J.R. Hughes.c© 2004 John Wiley & Sons, Ltd.

2 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS

on a range of length scales down to a smallest scale of size to Re−3/4, assuming L = 1. In manyapplications of scientific and industrial importanceRe is very large, of the order 106 or larger, and theflow shows a combination of laminar and turbulent features. To accurately resolve all the features of aflow at Re = 106 in a Direct Numerical Simulation DNS would require of the orderRe3 = 1018 meshpoints in space-time, and thus would be impossible on any forseeable computer.

To overcome the impossibility of DNS at higher Reynolds numbers various techniques of turbulencemodeling have been proposed. In a Large Eddy Simulation (LES) the coarser scales of the flow areresolved by the mesh and one seeks to model the influence of the unresolved small scales on theresolved larger scales in a subgrid model, see Sagaut, 2001, for an overview of LES and relatedtechniques. Various subgrid models have been proposed, but no clear answer to the question of thefeasibility of LES in simulation of turbulence has been given. A main open problem of CFD today isthe simulation of turbulent flow at high Reynolds numbers.

1.1. Adaptivity for laminar flow

In an adaptive algorithm, the computational method is adaptively modified during the computationbased on a modification strategy, until that a certain stopping criterion is satisfied. Error indicators andstopping criterions may be constructed ad hoc or using local truncation errors, see e.g. Berger et al.,1996, or being based on a posteriori error estimates of the global error.

A posteriori error estimation is traditionally done with respect to an energy-norm, naturally inducedby the underlying differential operator, resulting in estimates in terms of computable residuals, seeBabuska and Rheinboldt, 1978. For surveys and references on this approach we refer to Verfurth, 1996;Ainsworth and Oden, 1997. Unfortunately, in most applications the energy-norm does not provideuseful bounds on the error in quantities of real physical interest. Therefore, modifications of thisframework have been proposed for a posteriori error estimation also for linear functional output, seeOden and Prudhomme, 2002; Patera and Peraire, 2002; Bank and Welfert, 1991; Bank and Smith,1993.

Another approach is to use duality arguments to obtain bounds on the error in other norms, such asthe L2-norm, or the error in various functional output, such as drag or lift forces for example. The ideaof using duality arguments in a posteriori error estimation goes back to Babuska and Miller, 1984a;Babuska and Miller, 1984b; Babuska and Miller, 1984c, in the context of postprocessing ’quantitiesof physial interest’ in elliptic model problems. A framework for more general situations has sincethen been systematically developed Eriksson and Johnson, 1988; Eriksson et al., 1995; Becker andRannacher, 1996; Becker and Rannacher, 2001; Johnson and Rannacher, 1994; Johnson et al., 1995;Bank and Holst, 2003. For a detailed account of the development and application of this frameworkwe refer in particular to the review papers Eriksson et al., 1995; Becker and Rannacher, 2001; Gilesand Suli, 2002, and the references therin. A posteriori error estimation for hyperbolic problems andstabilized finite element methods have been investigated in Houston et al., 2000a. In CFD, applicationsof adaptive finite element methods based on this framework have been increasingly advanced, withcomputations of functionals such as drag and lift forces, for 2d and 3d laminar incompressible flow inBecker and Rannacher, 1996; Giles et al., 1997; Heuveline and Rannacher, 2003; Braack and Richter,2006, and laminar compressible flow in Johnson, 1998; Hartmann and Houston, 2002; Barth, 2003;Burman et al., 2003; Burman, 2000a; Burman, 2000b; Sandboge, 1998.

In general the adaptive methods concern adaptive refinement of the computational mesh in spaceand/or time, but also adaptive choice of the approximation order of the method is possible in thecontext of hp-methods, see e.g. Heuveline and Rannacher, 2003; Houston et al., 2000b.


COMPUTABILITY AND ADAPTIVITY IN CFD 3

1.2. Adaptive methods for LES

Less work on adaptivity is avaliable for turbulent flow. In Hoffman, 2004, a posteriori error analysisfor LES is developed, taking into consideration both the numerical error from discretization of thefiltered Navier-Stokes equations, and the modeling error from unresolved subgrid scales. Hoffman,2004, represents the first presentation of a posteriori error estimates with respect to a filtered solutionof the turbulent Navier-Stokes equations in 3d, of the form discretization and modeling residualstimes dual weights, with numerical approximation of the dual weights through computation of timedependent linearized dual Navier-Stokes equations in 3d. This approach to a posteriori error estimationwith respect to the averaged solution, using duality techniques, in terms of a modeling error and adiscretization error was developed for convection-diffusion-reaction equations in Hoffman et al., 2005;Hoffman, 2003; Hoffman, 2001.

Related approaches with a posteriori error estimates in terms of a modeling and a discretizationcontribution to the total error have been suggested. For example, in Braack and Ern, 2003, similarideas are presented with applications to 2d convection-diffusion-reaction problems, and in Ervin et al.,2000, the notion of a modeling error is present in a framework for defect-correction with applicationto the stationary Navier-Stokes equations in 2d, although the estimates bound the error with respect tothe unfiltered solution as opposed to the approach in Hoffman, 2004, where the error control is donewith respect to the filtered solution. A posteriori error estimation with respect to spatial averages inthe Navier-Stokes equations is considered in Dunca et al., 2004; dunca and John, 2004, where it isshown that for the stationary Navier-Stokes equations, under an assumption of strong stability for thecorresponding dual problem, the filtered error converges of higher order than the unfiltered error inL2-norm. In Dunca et al., 2003; Vasilyev et al., 1998, the commutation error of LES is investigated.

1.3. Adaptive computation of approximate weak solutions

We present in these notes our recent work in Hoffman and Johnson, 2006b; Hoffman, 2005a;Hoffman, 2006a; Hoffman, 2005b; Hoffman, 2005c; Hoffman, 2006b; Hoffman and Johnson, 2006a,on adaptivity and computability for turbulent flow, which is based on an extension of the frameworkfor adaptive stabilized Galerkin finite element methods with duality based a posteriori error controlfor chosen output quantities of interest. Our basic tool is a stabilized Galerkin method together withan a posteriori error estimate derived from a representation of the error in output in terms of a space-time integral of the residual of a computed solution multiplied by weights related to derivatives ofthe solution of an associated dual linearized problem with data connected to the output. We use anadaptive procedure where we compute on a sequence of successively refined meshes with the objectiveof reaching a stopping criterion based on the a posteriori error estimate with minimal computationaleffort (minimal number of mesh points in space-time). We show in concrete examples that outputs suchas a mean value in time of the drag of a bluff body in a turbulent flow is computable on a PC (withtolerances on the level of a few percent using a few hundred thousand mesh points in space).

The stabilized Galerkin method is the Galerkin/least squares space-time finite element methoddeveloped over the years together with Hughes, Tezduyar and coworkers, see e.g. Brooks and Hughes,1982; Tezduyar, 1992; Johnson and Saranen, 1986, here referred to as the General Galerkin (G2)method. This method includes the streamline diffusion method on Eulerian space-time meshes, thecharacteristic Galerkin method on Lagrangian space-time meshes with orientation along particletrajectories, and Arbitrary Lagrangian-Eulerian ALE methods with different mesh orientation. G2offers a general flexible methodology for the discretization of the incompressible and compressibleNavier-Stokes equations applicable to a great variety of flow problems from creeping low Reynolds



number flow through medium to large Reynolds number turbulent flow, including free or movingboundaries. With continuous piecewise polynomials in space of order p and discontinuous orcontinuous piecewise polynomials in time of order q, we refer to this method as cG(p)dG(q) orcG(p)cG(q). Below we present computational results with cG(1)cG(1).

We sometimes refer to G2 for CFD as Adaptive DNS/LES, where automatically by adaptivity certainfeatures of the flow are pointwise resolved in a DNS, while certain other small scale features are leftunderresolved in a LES with the stabilization of the method acting as a simple subgrid model. The aposteriori error estimate underlying the mesh modification strategy and the stopping criterion has (i) acontribution from the residual of the Galerkin solution inserted in the Navier-Stokes equations and (ii)a contribution from the stabilization. If we reach the stopping criterion, this means that the sum of (i)and (ii) are below the tolerance, and thus in particular that the contribution from the stabilization to theoutput error is below the tolerance.

We show in examples that certain outputs of a partly turbulent flow are computable without resolvingall the small scale features of the turbulent flow by using the simple subgrid model in the form of thestabilization of G2. The reason this is possible is that the output does not depend on all the exact detailsof the turbulent flow, which we observe by computing the contribution to the a posterori error estimatefrom the subgrid model and noting that this contribution indeed is small. This is analogous to theobserved fact that in a shock problem for compressible flow, certain outputs are computable withoutresolving the shocks to their actual width.

The key new ingredient in our work as presented in these notes is the use of duality to assess the basicproblem of computability of a specific output for a specific flow. In particular, we show that the crucialdual solution may be computed at an affordable cost for complex time-dependent laminar/turbulentflows in 3d.

We may view the stabilized Galerkin method as producing an approximate weak solution of theNavier Stokes equations, with a residual which in a weak sense is small (reflecting the Galerkinorthogonality) and which is also controlled in a strong sense (reflecting the least squares stabilization).

2. The incompressible Navier–Stokes equations

The Navier–Stokes equations were formulated in 1821–45 and seem to give an accurate description ofa great variety of fluid flows, including both laminar flow with ordered flow features and turbulent flowwith smaller and larger vortices in a complex interaction. Even though the Navier–Stokes equationshave been known for almost 200 years, a basic mathematical understanding of the equations is missing,and computational simulation of turbulent flow is still an outstanding open problem.

The incompressible Navier–Stokes (NS) equations for a unit density Newtonian fluid with constantkinematic viscosity ν > 0 enclosed in a volume Ω in R

3 (where we assume that Ω is a polygonaldomain), together with suitable boundary conditions, take the form:

R(u) = 0, in Ω × I, (1)

for u = (u, p), with u(x, t) the velocity vector and p(x, t) the pressure at (x, t), I = (0, T ) is a timeinterval, and R(u) ≡ R(u) − (f, 0) = (R1(u), R2(u)) − (f, 0), with

R1(u) = u+ u · ∇u+ ∇p− ν∆u,

R2(u) = ∇ · u. (2)



A given flow may be characterized by the Reynolds number Re = UL/ν, where U and L arerepresentative velocity and length scales, respectively, and with U and L normalized to unity we haveRe = ν−1. For Re larger than ∼ 100 we expect to have a partly turbulent flow, and to resolve all thephysical scales in a turbulent flow, one may estimate the number of mesh points needed to be of theorder Re3 in space-time, see e.g. Frisch et al., 1995. In many applications of industrial importance wehave Re > 106, and thus a Direct Numerical Simulation DNS of the NS equations is impossible.

Many attempts have been made to try to mathematically analyze the NS equations, but still the basicquestions of existence and uniqueness of solutions stand without a clear answer. A major contributionwas made by Jean Leray, who in 1934 proved the existence of a so called weak solution (or turbulentsolution in the terminology of Leray), see Leray, 1934, which is a solution satisfying (1) in an averagesense, so that u would be a weak solution if

((R(u), v)) = 0, (3)

for all test functions v in a test space V with norm ‖·‖V , consisting of suitable differentiable functions,and R(u) is assumed to belong to a space dual to V , and ((·, ·)) denotes a duality pairing.

An idea in contemporary mathematics research on the NS equations is now to try to extend theresult of Leray, to prove existence and uniqueness of a strong solution u, that would make the residualpointwise zero; R(u) = 0. The idea, expressed in the Clay Institue $1 million Prize Problem, seeFefferman, 2000, would be to prove that in fact Leray’s weak solution u is a strong solution, by provingregularity of u. With a pointwise bound on first order derivatives, uniqueness would then follow from astandard Gronwall estimate, bounding the pointwise difference between two strong solutions in termsof a small perturbation in data, which one then would let go to zero, which would give uniqueness.

Although, the use of Gronwall would bring in an exponential factor eKT , with K a pointwise boundof the gradient of the velocity, which may be estimated to be of the order K ∼ Re1/2 in a turbulentflow. For Re = 106 and T = 10, this would result in the enormous factor e10000, which is muchlarger than 10100, a googol. Any uniqueness proof involving such constants seems to lack any practicalmeaning, since virtually anything could then result from almost nothing.

The problem of proving uniqueness should come as no surprise, since in a turbulent flow we mayexpect an extreme pointwise perturbation growth, and thus pointwise uniqueness is too much to askfor. On the other hand, some aspects of turbulent flow are more well determined. Typically we expectvarious types of mean value output in a turbulent flow to be more well determined than pointwiseoutput. A basic question is then: what outputs of a turbulent flow are computable? We approach thisquestion in the following quantitative form:

• What outputs of a turbulent flow are computable to what tolerance to what cost?

Uniqueness in output of a solution is intimately connected to predictability and computability of suchan output, as they are all related to perturbation growth in the model, where predictability is related toerrors in output from data input, and computability is related to errors in output from computation.

In a turbulent flow we do not expect pointvalues to be predictable/computable to any tolerance ofinterest, but we expect that certain mean values can be predicted/computed to a tolerance of interest. Wemay compare to weather prediction, which concerns solving equations of NS type for the atmosphere:in January we may expect to be able to predict the mean temperature in London for the month of Juneto a tolerance of say ±2C, but we do not expect to be able to predict the daily temperature in LondonJune 20th to a tolerance of less than say ±10C, which is not very useful. For the problem of computingthe mean drag of bluff bodies in a turbulent flow, a tolerance of about 5-10% seems reasonable, based



on computational and experimental experience, but we cannot expect to be able to predict mean dragto an arbitrary small tolerance.

In Hoffman and Johnson, 2006a, we present a new approach to assess the uniqueness, predictability,and computability of various mean value output in turbulent flow, where we relax Leray’s weak solutionconcept, to the new concept of an ε-weak solution, where we say that u is an ε-weak solution if

|((R(u), v))| ≤ ε‖v‖V , ∀v ∈ V ,

where ε is a (small) positive number. This means that for an ε-weak solution u, we require the residualR(u) to be smaller than ε in a weak norm dual to the strong norm of V , and we denote the spaceof ε-weak solutions Wε. We note that this approach is in direct contrast to the approach taken in theformulation of the Clay Prize, seeking a proof that Leray’s weak solution is a strong solution: here werelax the requirements on the solution, and choosing ε = 0 would then bring us back to Leray’s weaksolution concept.

We show that for two ε-weak solutions u, w ∈Wε, we have that

|M(u) −M(w)| ≤ 2εSε(ψ), (4)

where Sε(ψ) is a stability factor characterizing the stability properties of the linear functional outputM(·), defined byM(w) ≡ ((w, ψ)). The estimate (4) expresses output uniqueness of ε-weak solutionsin a quantitative form, where the stability factor Sε(ψ) is determined by computational approximationof a dual problem, linearized at u and w, and with data ψ connecting to the outputM(·). The crucial factbehind the success of this approach for turbulent flow is that Sε(ψ) turns out to be stable with respect tolinearization at different elements inWε for mean value outputM(·), which we show computationally.

We thus relax the solution concept two-fold: (i) by using a weak solution concept, and (ii) byasking R(u) to be zero in a weak sense only approximatively. We shall see that (i) and (ii) naturallycome together; asking only (i) does not make much sense. With the standard Leray weak solution,corresponding to ε = 0, the stability information in (4) would be lost, since one may argue that anythingtimes 0 is still 0.

Further, we find that we are able to construct ε-weak solutions for almost any ε > 0, by usingstabilized Galerkin finite element methods, here referred to as General Galerkin G2 methods. For a G2solution U we derive a posteriori error estimates, with respect to an ε-weak solution u, of the form

|M(u) −M(U)| ≤ εSε(ψ) +∑

K∈T

eKD + eK

M , (5)

with error contributions from each elementK in the computational mesh T , where eKD is an error from

the Galerkin discretization in G2, eKM is an error contribution from the stabilization in G2, and εSε(ψ)

characterizes computability of the output M(·) in terms of a best possible accuracy, where ε is givenby the maximal computational resources.

One could also add a contribution εdataSεdata(ψ) to (5) from errors in data, where εdata characterizes

precision in data measurements. That is, given a certain disturbance level εdata in data (initial data,boundary data, etc.), the resulting variation in the output M(·) is bounded by εdataSεdata

(ψ). Forexample, experimental measurements of the drag coefficient of a circular cylinder at Re = 3900 ispresented as 0.98 ± 0.05, where the disturbance level in the experiments results in an output error ofabout 5%.

It seems reasonable to aim for an accuracy in computations similar to the one in experiments. Weconstruct an adaptive computational method based on (5) for the approximation of the outputM(·) to a



given accuracy TOL, using a minimal number of degrees of freedom, by formulating the minimizationproblem:

• Find a G2 solution U , with a minimal number of degrees of freedom, such that |M(u)−M(U)| ≤TOL,

which we solve by using the a posteriori error estimate (5), by starting from a coarse mesh andsuccessively adding new degrees of freedom based on (5) until the output error is less than the giventolerance TOL. For turbulent flow we sometimes refer to this method as Adaptive DNS/LES, with partsof the flow being resolved to it’s physical scales in a DNS, and parts of the flow being left unresolvedin a LES, with the stabilization in G2 acting as a subgrid model.

3. ε-Weak solutions to the Navier-Stokes equations

Viewing the study of the NS equations, in the general case of turbulent flow, as the study of meanvalue output from the equations, we do not need to worry about the possible existence of any strongsolutions. Instead we focus on the existence of approximate weak solutions, and (weak) uniqueness inoutput of such solutions.

To study weak uniqueness of the NS equations we introduce the concept of an ε-weak solution,which is an approximate weak solution with a residual less than ε in a weak norm. We define forv = (v, q) ∈ V ,

((R(u), v)) ≡ ((u, v)) + ((u · ∇u, v)) − ((∇ · v, p))+ ((∇ · u, q)) + ((ν∇u,∇v)) − ((f, v)),

(6)

whereV = v = (v, q) ∈ [H1(Q)]4 : v ∈ L2(H

10 (Ω))3,

and ((·, ·)) is the [L2(Q)]m inner product with m = 1, 3, or a suitable duality pairing, over the space-time domain Q = Ω × I . In order for all the terms in (6) to be defined, we ask u ∈ L2(I ;H

10 (Ω)3),

(u · ∇)u ∈ L2(I ;H−1(Ω)3), u ∈ L2(I ;H

−1(Ω)3), p ∈ L2(I ;L2(Ω)), and f ∈ L2(I ;H−1(Ω)3),

where H10 (Ω) is the usual Sobolev space of vector functions being zero on the boundary Γ and square

integrable together with their first derivatives over Ω, with dual H−1(Ω). As usual, L2(I ;X) with Xa Hilbert space denotes the set of functions v : I → X which are square integrable.

We now define u ∈ V to be an ε-weak solution if

|((R(u), v))| ≤ ε‖v‖V , ∀v ∈ V , (7)

where ‖ · ‖V denotes theH1(Q)4-norm, and we define Wε to be the set of ε-weak solutions for a givenε > 0. Note that for simplicity we here ask also the solution u to belong to the test space V , whichrequire more regularity than necessary; for the formulation (7) to make sense, it is sufficient that R(u)belongs the dual space of V . Equivalently, we may say that u ∈ V is an ε-weak solution if

‖R(u)‖V ′ ≤ ε,

where ‖ · ‖V ′ is the dual norm of V . This is a weak norm measuring mean values of R(u) withdecreasing weight as the size of the mean value decreases. Pointvalues are thus measured very lightly.



Formally we obtain the equation((R(u), v)) = 0

by multiplying the NS equation by v, that is, the momentum equation by v and the incompressibilityequation by q, and integrating in space-time. Thus, a pointwise solution u to the NS equations would bean ε-weak solution for all ε ≥ 0, while an ε-weak solution for ε > 0 may be viewed as an approximateweak solution.

Below, we show how to construct ε-weak solutions using stabilized finite element methods, and thusexistence of ε-weak solutions for any ε > 0 is guaranteed (under a certain assumption). For a computedsolution u, we can determine the corresponding ε by evaluating the residual R(u).

4. Output sensitivity and the dual problem

Suppose now the quantity of interest, or output, related to a given ε-weak solution u is a scalar quantityof the form

M(u) = ((u, ψ)), (8)

which represents a mean-value in space-time, where ψ ∈ L2(Q) is a given weight function. In typicalapplications the output could be a drag or lift coefficient in a bluff body problem. In this case theweight ψ is a piecewise constant in space-time. More generally, ψ may a piecewise smooth functioncorresponding to a mean-value output.

We now seek to estimate the difference in output of two different ε-weak solutions u = (u, p) andw = (w, r). We thus seek to estimate a certain form of output sensitivity of the space Wε of ε-weaksolutions. To this end, we introduce the following linearized dual problem of finding ϕ = (ϕ, θ) ∈ V ,such that

a(u, w; v, ϕ) = ((v, ψ)), ∀v ∈ V0, (9)

where V0 = v ∈ V : v(·, 0) = 0, and

a(u, w; v, ϕ) ≡ ((v, ϕ)) + ((u · ∇v, ϕ)) + ((v · ∇w,ϕ))

+ ((∇ · ϕ, q)) − ((∇ · v, θ)) + ((ν∇v,∇ϕ)),

with u and w acting as coefficients, and ψ is given data.This is a linear convection-diffusion-reaction problem in variational form, u acting as the convection

coefficient and ∇w as the reaction coefficient, and the time variable runs “backwards” in time withinitial value (ϕ(·, T ) = 0) given at final time T imposed by the variational formulation. The reactioncoefficient ∇w may be large and highly fluctuating, and the convection velocity u may also befluctuating.

Choosing now v = u− w in (9), we obtain

((u, ψ)) − ((w, ψ)) = a(u, w; u− w, ϕ) = ((R(u), ϕ)) − ((R(w), ϕ)), (10)

and thus we may estimate the difference in output as follows:

|M(u) −M(w)| ≤ 2ε‖ϕ‖V . (11)

By defining the stability factor S(u, w; ψ) = ‖ϕ‖V , we can write

|M(u) −M(w)| ≤ 2εS(u, w; ψ), (12)



and by definingSε(ψ) = sup

u,w∈Wε

S(u, w; ψ), (13)

we get|M(u) −M(w)| ≤ 2εSε(ψ), (14)

which expresses output uniqueness of Wε.Clearly, Sε(ψ) is a decreasing function of ε and we may expect Sε(ψ) to tend to a limit S0(ψ) as ε

tends to zero. For small ε, we thus expect to be able to simplify (14) to

|M(u) −M(w)| ≤ 2εS0(ψ). (15)

Depending on ψ, the stability factor S0(ψ) may be small, medium, or large, reflecting differentlevels of output sensitivity, where we expect S0(ψ) to increase as the mean value becomes more local.Normalizing, we may expect the output M(u) ∼ 1, and then one would need 2εS0(ψ) < 1 in orderfor two ε-weak solutions to have a similar output.

Estimating S0(ψ) using a standard Gronwall type estimate of the solution ϕ in terms of the data ψwould give a bound of the form S0(ψ) ≤ CeKT , whereC is a constant and andK is a pointwise boundof |∇w|. In a turbulent flow with Re = 106 we may have K ∼ 103, and with T = 10 such a Gronwallupper bound of S0(ψ) would be of the form S0(ψ) ≤ CeKT ∼ e10000, which is an incredibly largenumber, larger than 10100, a googol. It would be inconceivable to have ε < 10−100, and thus the outputof an ε-weak solution would not seem to be well defined.

However, computing the dual solution corresponding to drag and lift coefficients in turbulent flow,we find values of S0(ψ) which are much smaller, in the range S0(ψ) ≈ 102 − 103, for which it ispossible to choose ε so that 2εS0(ψ) < 1, with the corresponding outputs thus being well defined(up to a certain tolerance). In practice, there is a lower limit for ε, typically given by the maximalcomputational cost, and thus S0(ψ) effectively determines the computability of different outputs.

We attribute the fact that ϕ and derivatives thereof are not exponentially large, to cancellation effectsfrom the oscillating reaction coefficient ∇w, with the sum of the real parts of the eigenvalues of ∇wbeing very small, with the sign being about as often positive as negative, see Fig. 5. These cancellationeffects seem to be impossible to account for in purely theoretical estimates, because knowledge of theunderlying flow velocity is necessary.

5. Existence of ε-weak solutions

To generate approximate weak solutions we use a stabilized finite element method of the form: FindU ≡ Uh ∈ Vh, where Vh ⊂ V is a finite dimensional subspace defined on a computational mesh inspace-time of mesh size h, such that

((R(U), v)) + ((hR(U), R(v))) = 0, ∀v ∈ Vh, (16)

where R(U) ≡ R(U) − (f, 0) = (R1(U), R2(U)) − (f, 0), and for w = (w, r),

R1(w) = w + U · ∇w + ∇r − ν∆w,

R2(w) = ∇ · w, (17)



with elementwise definition of second order terms. We here interpret a convection term ((U · ∇w, v))as

1

2((U · ∇w, v)) − 1

2((U · ∇v, w),

which is literally true if ∇ · U = 0. With this interpretation we will have ((U · ∇U,U)) = 0, even ifthe divergence of the finite element velocity U does not vanish exactly, and we obtain choosing v = Uin (16), and assuming that f = 0:

((ν∇U,∇U)) + ((hR(U ), R(U))) ≤ ‖u0‖2. (18)

The finite element method (16) is a stabilized Galerkin method with the term ((R(U), v))corresponding to Galerkin’s method and the term ((hR(u), R(v))) corresponding to a weightedresidual least squares method with stabilizing effect expressed in (18). We also refer to this methodas G2 or General Galerkin, and we thus refer to U as a G2 solution. The existence of a discrete solutionU ≡ Uh ∈ Vh follows by Brouwer’s fixed point theorem combined with the stability estimate (18).

We now prove the existence of ε-weak solutions to the NS equations for any ε > 0. For all v ∈ V ,we have with vh ∈ Vh a standard interpolant of v satisfying ‖h−1(v − vh)‖ ≤ Ci‖v‖V , using also(16),

|((R(U), v))| = |((R(U ), v − vh)) − ((hR(U), R(vh))|≤ Ci‖hR(U)‖‖v‖V +M(U)‖hR(U)‖‖v‖V ,

(19)

where M(U) is a pointwise bound of the velocity U(x, t), and Ci ≈ 1 is an interpolation constant. Itfollows that the G2-solution U is an ε-weak solution with

ε = (Ci +M(U))‖hR(U)‖ ≤√h(Ci +M(U))‖u0‖,

since from the energy stability estimate (18) we have that ‖√hR(U)‖ ≤ ‖u0‖.

Assuming now that M(U) = M(Uh) is bounded with h > 0, and letting (Ci +M(U))‖u0‖ ≤ C,it follows that U is an ε-weak solution with ε = C

√h. More generally, we may say that a G2 solution

U is an ε-weak solution with ε = CU‖hR(U)‖, with CU = Ci +M(U).We have now demonstrated the existence of an ε-weak solution to the NS equations for any ε,

assuming that the maximum of the computed velocity is bounded (or grows slower than h−1/2). Moregenerally, we have shown that a G2 solution U is an ε-weak solution with ε = CU‖hR(U)‖. ComputingU , we can compute ε = CU‖hR(U)‖ and thus determine the corresponding ε.

6. Computability and a posteriori error estimation for G2

We now let u be an ε-weak solution of the NS equations with ε small, and we let U be a G2 solution,which can be viewed to be an εG2-weak solution, with εG2 = CU‖hR(U)‖ >> ε. As in (14), we getthe following a posteriori error estimate for a mean value output given by a function ψ:

|M(u) −M(U)| ≤ (CU‖hR(U)‖ + ε)SεG2(ψ), (20)

where SεG2(ψ) is the corresponing stability factor defined by (13). Obviously the size of the stability

factor SεG2(ψ) is crucial for computability: the stopping criterion is evidently (assuming ε small):

CU‖hR(U)‖SεG2(ψ) ≤ TOL,



where TOL > 0 is a tolerance. If SεG2(ψ) is too large, or TOL is too small, then we may not be able

to reach the stopping criterion with available computing power, and the computability is out of reach.In applications we estimate SεG2

by computional approximation of the dual problem.We note that for weak uniqueness the residual only needs to be small in a weak norm, and

correspondingly for computability the G2 residual only needs to be small when weighted by h.This means that for accurate approximation of a mean value output, the NS equations do not needto be satisfied pointwise, corresponding to a pointwise small residual, but only in an average sense,corresponding to the residual being small only in a weak norm. In computations we find that in fact theG2 residual typically is large pointwise for solutions corresponding to accurate approximation of meanvalue output, such as the drag of a bluff body.

7. The Eulerian cG(1)cG(1) method

The cG(1)cG(1) method is a variant of G2 using the continuous Galerkin method cG(1) in space andtime. With cG(1) in time the trial functions are continuous piecewise linear and the test functionspiecewise constant. cG(1) in space corresponds to both test functions and trial functions beingcontinuous piecewise linear. Let 0 = t0 < t1 < ... < tN = T be a sequence of discrete timesteps with associated time intervals In = (tn−1, tn] of length kn = tn − tn−1 and space-time slabsSn = Ω × In, and let W n ⊂ H1(Ω) be a finite element space consisting of continuous piecewiselinear functions on a mesh Tn = K of mesh size hn(x) with Wn

w the functions v ∈ W n satisfyingthe Dirichlet boundary condition v|∂Ω = w.

We now seek functions U = (U, P ), continuous piecewise linear in space and time, and thecG(1)cG(1) method for the Navier-Stokes equations (1), with homogeneous Dirichlet boundaryconditions reads: For n = 1, ..., N , find (Un, Pn) ≡ (U(tn), P (tn)) with Un ∈ V n

0 ≡ [Wn0 ]3 and

Pn ∈Wn, such that

((Un − Un−1)k−1n + Un · ∇Un, v) + (2νε(Un), ε(v)) − (P n,∇ · v)

+ (∇ · Un, q) + SDδ(Un, Pn; v, q) = (f, v) ∀(v, q) ∈ V n

0 ×Wn,(21)

where Un = 1

2(Un + Un−1), with the stabilizing term

SDδ(Un, Pn; v, q) ≡ (δ1(U

n · ∇Un + ∇Pn − f), Un · ∇v + ∇q) + (δ2∇ · Un,∇ · v),

with δ1 = 1

2(k−2

n + |U |2h−2n )−1/2 in the convection-dominated case ν < Unhn and δ1 = κ1h

2n

otherwise, δ2 = κ2hn if ν < Unhn and δ2 = κ2h2n otherwise, with κ1 and κ2 positive constants of

unit size (here we have κ1 = κ2 = 1), and

(v, w) =∑

K∈Tn

∫

K

v · w dx,

(ε(v), ε(w)) =3

∑

i,j=1

(εij(v), εij(w)).

We note that the time step kn is given by the mesh size hn, with typically

kn ∼ minxhn(x)/Un(x).



8. Adaptive algorithm

A mean value in time of the force on a body, over a time interval I , may be expressed as

N(σ(u)) =1

|I |

∫

I

(u+ u · ∇u− f,Φ) − (p,∇ · Φ)

+(2νε(u), ε(Φ)) + (∇ · u,Θ) dt, (22)

where u is an ε-weak solution to the NS equations, and Φ is a function defined in the fluid volume Ωbeing equal to a unit vector in the direction of the force we want to compute on Γ0, the surface of thebody in contact with the fluid, and zero on the remaining part of the boundary Γ1 = ∂Ω \ Γ0. Therepresentation (22) is independent of Θ, and the particular extension of Φ away from the boundary, andwe ask that Φ = (Φ,Θ) ∈ V .

We compute an approximation of the dragN(σ(u)) from a cG(1)cG(1) solution U , using the formula

Nh(σ(U )) =1

|I |

∫

I

(U + U · ∇U − f,Φ) − (P,∇ · Φ)

+(2νε(U), ε(Φ)) + (∇ · U,Θ) + SDδ(U, P ; Φ,Θ) dt, (23)

where now Φ and Θ are finite element functions, and where U = (Un −Un−1)/kn on In. We note thepresence of the stabilizing term SDδ in (23) compared to (22), which is added in order to obtain theindependence of Nh(σ(U )) from the choice of (Φ,Θ), given by (21).

Approximating ϕ = (ϕ, θ), the exact solution to the dual problem (9), by a computed approximationϕh = (ϕh, θh), with the linearized convection velocity u ≈ U , we are led to the following a posteriorierror estimate, see Hoffman, 2005a, for the time average over I of the drag force on a body in a fluid,with respect to u ∈ Wε:

|N(σ(u)) −Nh(σ(U ))| ≈ εSε(ψ) +∑

K∈Tn

EK,h, (24)

where ψ = (Φ, 0) is the data to the dual problem defining the output N(σ(·)), k and h are the timestep and the local mesh size, respectively, and EK,h = eK

D,h + eKM,h, with

eKD,h =

1

|I |

∫

I

(

(|R1(U)|K + |R2(U)|K) · (Chh2|D2ϕh|K + Ckk|ϕh|K)

+ ‖R3(U)‖K · (Chh2‖D2θh‖K + Ckk‖θh‖K)

)

dt,

eKM,h =

1

|I |

∫

I

SDδ(U ; ϕh)K dt,

where we may view eKD,h as the error contribution from the Galerkin discretization in cG(1)cG(1), and

eKM,h as the contribution from the stabilization in cG(1)cG(1), on element K. The lower bound on the

tolerance, defining computability of the output N(σ(·)), is given by εSε(ψ). Here we think of ε asbeing small, corresponding to a maximal computational cost, so that εSε(ψ) <<

∑

K∈TnEK,h.

We now present an algorithm for adaptive mesh refinement based on the a posteriori output errorestimate (24). For simplicity, we here use the same space mesh and the same time step length for alltime steps.

Given an initial coarse computational space mesh T 0, start at k = 0, then do



(1) Compute approximation of the primal problem using T k.(2) Compute approximation of the dual problem using T k.(3) If |

∑

K∈Tk

EkK,h| < TOL then STOP, else:

(4) Based on the size of the local error indicator EkK,h, mark a fixed fraction of the elements in T k

for refinement. Obtain a new refined mesh T k+1, using a standard algorithm for local meshrefinement.

(5) Set k = k + 1, then goto (1).

To assess the efficiency of G2, we now present computational results using G2 for a number of welldefined benchmark problems. We find that we are able to compute mean value output, such as forceson bluff bodies, using less than 105 mesh points in space, which is a very low number compared totypical LES computations using ad hoc mesh refinement. We further illustrate some aspects of thecorresponding dual problems, connecting to computability of mean value output in turbulent flow.

9. Benchmark problems

We now present results from Hoffman, 2005a; Hoffman, 2005b; Hoffman, 2006a, with G2 for a numberof benchmark problems, where we approximate a time average of a normalized force on an object in aturbulent flow We compare the results to LES computations in the literature, e.g. Zdravkovich, 1997;Schlichting, 1955; Kravchenko and Moin, 2000; Breuer, 1998; Mittal, 1996; Rodi et al., 1997; Buijssenand Turek, 2004; Krajnovic and Davidson, 2002. The idea of G2 is to compute a certain output to agiven tolerance, using a minimal number of degrees of freedom. We are thus interested in comparingthe number of mesh points needed in G2 to compute a certain output, compared to the number of meshpoints needed in other approaches.

The drag coefficient cD is defined as a global average of a normalized drag force on an object fromthe fluid. We seek to approximate the drag coefficient cD by cD, a normalized drag force averaged overa finite time interval I = [0, T ], at fully developed flow, defined by

cD ≡ 11

2U2∞A

×N(σ(u)), (25)

whereU∞ = 1 is a reference velocity,A is the projected area of the object facing the flow, andN(σ(u))is defined by (22) with Φ in the direction of the mean flow. In computations we approximate cD by chD,defined by

chD =1

1

2U2∞A

×Nh(σ(U )), (26)

with Nh(σ(U)) defined by (23). Thus we can use a scaled version of the a posteriori error estimate(24) to estimate the error |cD − chD|, upon which we base the adaptive algorithm.

Similarly, we define the lift coefficient cL as a global average of a normalized lift force on an objectfrom the flow, with analogous definitions of cL and chL, and where Φ is now in a direction perpendicularto the mean flow.



9.1. Flow past a circular cylinder

The turbulent flow past a circular cylinder at Re = 3900 is a well documentet problem, with manyexperimental and computational reference results avaliable, see e.g. Zdravkovich, 1997; Schlichting,1955; Kravchenko and Moin, 2000; Breuer, 1998; Mittal, 1996. An approximate solution using G2 isplotted in Fig. 1, where we observe vortex shedding and a large turbulent wake attached to the cylinder.

In Fig. 2, we plot computational approximations of the drag coefficient as we refine the mesh,refining 5% of the elements in each iteration of the adaptive algorithm. We have ch

D ≈ 0.97 for thefinest mesh, and we note that chD is within the experimental tolerance cD = 0.98± 0.05 for all meshesusing 28 831 nodes or more. We also capture the Strouhal number St ≈ 0.22, and the base pressurecoefficient CPb

= −0.90, within experimental tolerances. Thus using less than 105 mesh points inspace we are able to capture the correct cD, CPb

, and St, and using less than 30 000 mesh points weare able to capture cD within the experimental tolerance.

In Fig. 3 we plot snapshots of a dual solution corresponding to approximation of mean drag, and theresulting adaptively refined mesh. We find that the adaptive method automatically, by the sensitivityinformation in the dual problem, resolves the critical parts of the domain for accurate approximationof mean drag. Mesh refinement is concentrated to the turbulent wake, and the boundary layer of thecylinder to capture the correct separation points of the flow. New nodes are only added where they areneeded, which makes the method very efficient.

In Fig. 2 we plot the a posteriori error estimates of the error in cD, where we find that whereas themodeling error is steadily reduced as we refine the mesh, the discretization error is decreasing veryslowly from when we have reached the targeted value within the experimental tolerance 0.98± 0.05.

9.2. Flow past a sphere

The experiences from computing the turbulent flow past a sphere at Re = 104 using G2 is verymuch the same as for the circular cylinder. We plot the solution in Fig. 1, and in Fig. 2 we plot theapproximation of cD as we refine 5% of the elements in each iteration of the adaptive algorithm.Again using less than 30 000 nodes we capture the experimental reference value cD ≈ 0.40, see e.g.Schlichting, 1955; Constantinescu et al., 2002; Constantinescu et al., 2003, and for the finer meshesusing less than 105 nodes we capture the correct frequency St ≈ 0.20.

In Fig. 3 we plot snapshots of a dual solution and the resulting computational mesh, where we notethat again mesh refinement is concentrated to boundary layers and the turbulent wake.

9.3. Flow around a surface mounted cube

For the flow around a surface mounted cube at Reynolds number 40 000, the incoming flow is laminartime-independent with separation and a horse-shoe vortex upstream the cube, and a laminar boundarylayer on the front surface of the body, which separates and develops a turbulent time-dependent wakeattaching to the rear of the body, see Fig. 1.

In Figure 2 we plot approximations of cD as we refine the mesh, refining about 30% of the elementsin each iteration. We find that for the finer meshes ch

D ≈ 1.5, a value that is well captured alreadyusing less than 105 mesh points. Without the contribution from the stabilizing terms in (23), we get asomewhat lower value for chD, where the difference is large on coarser meshes but small (less than 5%)on the finer meshes.

We note that the convergence to the stable value is slower than for the cylinder and the sphere. Thisis due to the larger fraction of the elements being refined each iteration of the adaptive algorithm.



Figure 1. Out of paper vorticity for a sphere (upper left), streamwise vorticity component for a circular cylinder(upper right), pressure and negative pressure isosurfaces (lower left) and magnitude of the velocity (lower right)

for a surface mounted cube.

Refining less elements each iteration is more efficient in the sense that unnecessary refinement is morelikely avoided, but on the other hand we risk unnecessary many iterations.

9.4. Flow past a square cylinder

For a square cylinder at Re = 22 000 we have no separation of the flow upstream the cylinder, whichresults in higher pressure on the upstream face of the cylinder compared to the cube, and thus also ahigher drag cD ≈ 2.0, compared to cD ≈ 1.5. We reach the targeted value for the drag coefficientusing less than 105 mesh points in space, and thus again we have a significant cut of the computationalcost.



4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 50.8

1

1.2

1.4

1.6

1.8

2

3.8 4 4.2 4.4 4.6 4.80.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.5 3 3.5 4 4.5 5 5.50.5

1

1.5

2

2.5

3

3.5

4

4.5

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5

−1

−0.8

−0.6

−0.4

−0.2

0

Figure 2. chD with (’-’) and without (’:’) the contribution from the stabilizing term in (23) vs log10

number ofnodes; for a circular cylinder (upper left), a sphere (upper right), and for a surface mounted cube (lower left), and aposteriori error estimates from (24):

P

K∈TeK

D,h (’o’) andP

K∈TeK

M,h (’x’) for a circular cylinder (lower right).

10. Stability of the dual problem

The a posteriori error estimates underlying G2 are based on dual weights, which are approximated bycomputing solutions to the dual problem (9), linearized at a G2 solution U . A fundamental questionis now the stability of the dual weights, and thus the dual solution, with respect to perturbations from(i) computation, and (ii) from the linearization error u ≈ U in the convection velocity. In addition, tominimize the memory requirements we do not store the complete time series of the G2 solution, butwe use a quite coarse time sampling of U .

We now study the effect of the time sampling of U on the dual solution. Comparing differentsampling frequencies for the surface mounted cube problem, we find that the dual solution is quitestable with respect to the sampling frequency, see Fig. 4 and Tab. I, and thus a coarse samplingfrequency can be sufficient. Although, even though the differences are very small (less than 5%), wefind that there is a trend where a higher sampling frequency of U corresponds to a smaller dual solution,which may be explained by a higher cancellation in the dual problem linearized at coefficients with ahigh time variation.

We may further regard this as a test for stability of the dual solution with respect to linearizationat different G2 solutions, and we thus conclude that for this problem the dual solution is stable underperturbations from the linearization error in the coefficients of the dual problem.



Figure 3. Dual velocities |ϕh| and corresponding adapted meshes: for a circular cylinder at Re = 3900 (upper 2plots), and a sphere at Re = 104 (lower plot).

freq. ‖ϕh‖ ‖∇ϕh‖ ‖ϕh‖V

8 7.99 646 6524 8.04 658 6634∗ 8.14 679 6842 8.23 693 6981 8.35 744 749

Table I. Surface mounted cube: norms of the dual solution linearized at a G2 solution sampled with differentfrequencies, with “4∗” being a translated sampling of “4” with the same frequency.



Figure 4. Surface mounted cube: snapshots of the dual velocity |ϕh| linearized at a G2 solution U sampled 4 timesper time unit (upper left), 2 times per time unit (upper right), and once per time unit (lower left), corresponding to

“4”, “2”, and “1” in Table I, and an adaptively refined mesh based on “4” (lower right).

11. Weak uniqueness by computationWe now return to the problem of determining computability of an output in a turbulent flow, couplingto the problem of uniqueness in output of ε-weak solutions to the NS equations, or weak uniquenessfor short.

Weak uniqueness with respect to an output M(·) is expressed by (15), and in practise we seekto compute S0(ψ) approximately, replacing both u and w as coefficients in the dual problem bya computed ε-weak solution U , which may be a G2 solution, thus obtaining an approximate dualvelocity ϕh. We then study Sh(U ; ψ) ≡ ‖ϕh‖V as we refine the mesh size h, and we extrapolate toh = ν, representing a smallest scale. If the extrapolated value Sν(U ; ψ) < ∞, or rather is not toolarge, then we have evidence of weak uniqueness. If the extrapolated value is very large, we get anindication of weak non-uniqueness. As a crude test of largeness of Sν(U ; ψ) it appears natural to useSν(U ; ψ) >> ν−1/2 (as an estimate of the maximal size of the gradient, and thus the residual, basedon the basic energy estimate for the NS equations).

The dual problem is a convection-diffusion-reaction problem with the gradient of the G2 solutionacting as coefficient in the reaction term. Thus, for a turbulent flow the dual problem may seem verydangerous, with possible exponential growth from the reaction term. Although, it turns out that the G2gradient has a very particular structure: by the (approximate) incompressibility of the G2 solution, andthe fact that the sum of the eigenvalues of a matrix is equal to the sum of its diagonal elements, wehave that the sum of the real parts of the eigenvalues is very small, with the sign being about as often



positive as negative, see Fig. 5, resulting in cancellation and a very weak net growth of the crucialquantity Sh(U ; ψ), see Fig. 6.

200 400 600 800 1000 1200 1400 1600 1800−6

−4

−2

0

2

4

6

0 500 1000 1500 2000 2500 3000 3500−5

−4

−3

−2

−1

0

1

2

3

4

Figure 5. Sum of the real parts of the eigenvalues of ∇U , the jacobian of a G2 solution, for a few thousandelements in the turbulent wake of a circular (left) and a square cylinder (right).

We may use a slow growth of Sh(U ; ψ) as further evidence supporting that it is possible to replaceboth u and w by U in the computation of the dual problem: a near constancy for the finer meshesindicates a desired robustness to (possibly large) perturbations of the coefficients u and w.

We consider the computational example of the surface mounted cube presented above. In Figure 6we plot Sh(U ; ψ) corresponding to computation of cD as a function of h−1, with h the smallest elementdiameter in the mesh. We find that Sh(U ; ψ) shows a slow logarithmic growth, and extrapolating wefind that Sν(U ; ψ) ∼ ν−1/2. We take this as evidence of computability and weak uniqueness of cD .

We next investigate the computability and weak uniqueness of the normalized drag force D(t) ata specific time t in the time interval I = [0, T ]. In Figure 2 we show the variation in time of D(t)computed on different meshes, and we notice that even though the mean value cD has stabilized, D(t)for a given t appears to converge very slowly or not at all with decreasing h.

We now choose one of the finer meshes corresponding to h−1 ≈ 500, where we compute the dualsolution corresponding to a mean value of D(t) over a time interval [T0, T ], where we let T0 → T . Wethus seek to computeD(T ).

In Figure 6 we find a growth of Sh(U ; ψ) similar to |T − T0|−1/2, as we let T0 → T . The resultsshow that for |T − T0| = 1/16 we have Sh(U ; ψ) ≈ 10ν−1, and extrapolation of the computationalresults indicate further growth of Sh(U ; ψ), as T0 → T and h → ν. We take this as evidence ofnon-computability and weak non-uniqueness of D(T ).

We have thus given computational evidence of weak uniqueness of the mean value cD, and weaknon-uniqueness of a momentary value D(t) of the normalized drag force. In the computations weobserve this phenomenon as a continuous degradation of computability (increasing stability factorSh(U ; ψ)) as the length of the time interval underlying the mean value decreases to zero. Thecomputational cost sets a lower limit on ε for the ε-weak solution U . When the stability factor Sh(U ; ψ)is larger than ε−1, then we have effectively lost computability, since the oscillation of D(t) is of unitsize.

We compute cD as a mean value of finite length (of size 10), and thus we expect some variationalso in these values, but on a smaller scale than for D(t), maybe of size = 0.1 with 0.01 as a possiblelower limit with present computers. Thus the distinction between computability (or weak uniqueness)



2 2.5 3 3.5 4 4.5 5 5.5 60

0.5

1

1.5

2

2.5

0.8 1 1.2 1.4 1.6 1.8 2 2.2

2

2.2

2.4

2.6

2.8

3

2 2.5 3 3.5 4 4.5 5 5.5 60

500

1000

1500

2000

2500

3000

−1.5 −1 −0.5 0 0.5 12.8

2.9

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

Figure 6. Surface mounted cube: time series (with the time running backwards) of ‖∇ϕh‖ corresponding to meandrag cD for the 3 finest meshes (upper left), and log

10Sh(U ; ψ) as a function of log

10h−1 (upper right), time

series (with the time running backwards) of ‖∇ϕh‖ corresponding to computation of the (normalized) drag forceD(t) averaged over a time interval [T0, T ] for time intervals [T0, T ] = 4, 2, 1, 0.5 (lower left), and log

10Sh(U ; ψ)

as a function of the interval length log10

|T − T0| (lower right).

and non-computability (weak non-uniqueness) may in practice be just one or two orders of magnitudein output error, rather than a difference between 0 and ∞. Of course, this is what you may expect in aquantified computational world, as compared to an ideal mathematical world.

12. Rotating cylinder on moving ground

In Hoffman, 2005c, we consider the problem of computing the flow due to a cylinder rolling alongground. Computational studies investigating the influence of the contact with the ground, which isextremely important in many applications, seems to be largely lacking. In Bhattacharyya et al., 2004,this problem is investigated for a computational model in 2d for laminar flow, and here we consider amodel in 3d for turbulent flow. The main motivation comes from the automotive industry, where theflow of air past the wheels of a car or other vehicles is of much concern, since the drag of the wheels isa significant part of the total drag.

In our computational model we assume uniform rotation of a circular cylinder on flat ground, withthe length of the cylinder being equal to it’s diameter. In a coordinate frame moving with the constant



speed of the centre of the cylinder, the problem is to determine the flow past a uniformly rotatingcircular cylinder with a fixed centre and in contact with the ground moving with the same velocity asthe oncoming free stream of the fluid. The Reynolds number based on the free stream velocity and thecylinder diameter is set to Re = 104.

In Fig. 7 we plot the solution, where we compare the computational results for the rotating cylinderto the results a the stationary cylinder. One main difference between the two solutions is that for thestationary cylinder the flow separates to form a horse-shoe vortex upstream the cylinder, but for therotating cylinder we have no separation upstream. For separation to occur we need both a positivepressure gradient, resulting in a negative force in the momentum equation, and a boundary layer. Sincethe ground is moving with the same speed as the flow for the rotating cylinder, we have no boundarylayer, and thus no separation of the flow upstream the cylinder is possible. On the other hand, for thestationary cylinder we have a ground boundary layer leading to separation.

Under the cylinder close to the ground contact line, a lot of vorticity is produced for the rotatingcylinder, whereas this production appears to be less significant for the stationary cylinder. We also findthat the rotation of the cylinder makes the flow separate earlier than in the case of the non-rotatingcylinder, which makes the wake volume, and thus the drag, to increase. In addition, the wake volumeis also wider in the transversal direction, for the rotating cylinder.

The two flow configurations are thus fundamentally different, resulting in drag coefficients differingwith almost a factor 2; for the rotating cylinder cD ≈ 1.3, to compare with cD ≈ 0.8 for the stationarycylinder, see Fig. 8. This difference has important practical implications, such as the fact that the flowaround a stationary car in a wind tunnel may be very different from the flow around a moving car.

13. Drag crisis

The drag of a bluff body may be divided into pressure drag, coupling to the pressure drop over thebody, and skin friction. Separation of the flow influence the relation between pressure drag and skinfriction, with typically an earlier separation leading to higher pressure drag and lower skin friction.

For square geometries the separation is given by the geometry, see e.g. Hoffman, 2005a; Rodiet al., 1997; Krajnovic and Davidson, 2002, whereas for the flow past a circular geometry, theseparation is not given by the geometry, but depend on the Reynolds number, see e.g. Zdravkovich,1997; Schlichting, 1955; Kravchenko and Moin, 2000; Breuer, 1998; Mittal, 1996; Hoffman, 2005b;Hoffman, 2006a.

For a circular geometry, such as a circular cylinder or a sphere, the flow does not separate at allfor very low Reynolds numbers, where the skin friction dominates the contribution to the total drag,and then for increasing Reynolds numbers the wake increases and the separation points (lines) moveupstream until reaching θ ≈ 90, with θ an angle in the circular section counted from zero at theupstream stagnation point. In this configuration pressure drag is dominating skin friction, and the totaldrag is more or less constant (cD ≈ 1.0 for a circular cylinder and cD ≈ 0.4 for a sphere) for a widerange of Reynolds numbers (Re ≈ 103 − 105).

At very high Reynolds numbers (Re ≈ 105 − 106) the boundary layer for a circular cylinder anda sphere undergoes transition to turbulence, making the separation points (lines) move downstreamas the Reynolds number increases, resulting in a smaller wake and lower drag, so called drag crisis.Simulation of drag crisis is a major challenge of turbulence simulation.



Figure 7. Snapshots of magnitude of the vorticity for the rotating (left) and the stationary (right) cylinder.

13.1. Simulation of drag crisis

To resolve the very thin boundary layer of a high Reynolds number flow is too expensive, and thus manydifferent wall-models have been proposed to capture the effect of the boundary layer without resolvingit to its physical scale, see Sagaut, 2001, for an overview. In Constantinescu and Squires, 2004, aDetached-eddy simulation is used to simulate drag crisis for a sphere. A Detached-eddy simulation isa hybrid approach that has the behaviour of a RANS model near the boundary and becomes a LES inthe regions away from solid surfaces, see Sagaut, 2001.

In Hoffman, 2006b we propose a simple approach to model the effect of the unresolved turbulentboundary layer, based on a slip with friction boundary condition, which may be viewed as a verysimple wall-model. The idea is that the main effect of the boundary layer on the flow is skin friction.The problem is then to choose a suitable friction coefficient β (possibly varying in space and time).

In John, 2002; John and Liakos, 2005; John et al., 2003; Iliescu et al., 2002, similar boundary



4 4.2 4.4 4.6 4.8 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Figure 8. chD vs log10

# nodes for the rotating (’*’) and the stationary (’o’) cylinder.

conditions are used to study reattachement of a low Reynolds number flow past a surface mountedcube in 2d and 3d as a function of the friction parameter β, and it is found that reattachement isdelayed with decreasig friction, as expected.

For very high Reynolds numbers the viscous ν-term in the computational method (21) is neglible ifwe do not resolve the finest scales of the flow, and may be dropped from the equation, correspondingto a cG(1)cG(1) method for the Euler equations.

The criterion for choosing β should be that the skin friction in the computation should be the sameas in the physical problem. Experimental results, see e.g. Schlichting, 1955, indicate that skin frictionhas a very weak dependence on the Reynolds number, proportional toRe−0.2 in the case of a flat plate,and thus a certain value for β should be characteristic for a rather wide range of Reynolds numbers.Experimental results also indicate that once we have drag crisis the separation is again rather stablefor a range of Reynolds numbers. The exact Reynolds number for when the separation point starts tomove downstream seems to be hard to determine, which is probably due to its relation to the transitionto turbulence in the laminar boundary layers, which in turn depends on the level of perturbations inthe boundary layer, which is very hard to determine in a realistic problem. Thus, there is a range ofReynolds numbers, close to where transition in the boundary layers occur, for which the separation ofthe flow is very hard to determine. From an engineering point of view it is then important to take boththe sub-critical and the super-critical scenario into account.



Figure 9. Drag crisis: magnitude of the velocity for G2 solutions for ν = 0; β = 2 × 10−2 with cD ≈ 0.7 (upperleft), β = 1 × 10−2 with cD ≈ 0.5 (upper right), β = 5 × 10−3 with cD ≈ 0.45 (lower left), and the mesh with

80 000 mesh points (lower right), in the x1x2-plane for x3 = 0.2.

13.2. Drag crisis for a circular cylinder

In Hoffman, 2006b, we consider the problem of modeling drag crisis for the circular cylinder fromSection 9. Our model is here a cG(1)cG(1) method for the Euler equations together with a slip withfriction boundary condition. Letting the constant friction parameter β go from large to small values, wefind that the separation point is moving downstream. For β ≈ 10−2 we find that we are able to capturethe delayed separation of drag crisis with cD ≈ 0.4− 0.5, see Fig 9.



14. Turbulent Euler solutions

We now turn to the question of what happens as β → 0, corresponding to the Reynolds numberRe → ∞. Our computational model then reduces to G2 for the Euler equations with slip boundaryconditions, which we here refer to as an Euler/G2 model, or an EG2 model. We note that in this modelthe only parameter is the discretization parameter h.

Due to the absence of a boundary layer for a EG2 solution, we avoid using velocity boundary datafor the dual problem. Instead in Hoffman, 2006b, we introduce a modified dual problem with pressuredata, more suitable for this situation.

14.1. EG2 for a circular cylinder

Experimental results for the circular cylinder are avaliable up to Re ≈ 107, see e.g. Schlichting,1955; Zdravkovich, 1997, for which drag is small, corresponding to drag crisis. In wind tunnels thereis an upper limit on the size of the cylinder, and increasing the velocity eventually will make theincompressible flow model invalid, due to effects of compressibility and cavitation. To find muchhigher Reynolds numbers we have to consider flow problems with very large dimensions, such asgeophysical flows. In the recent book Zdravkovich, 1997, the case when Re→ ∞ is referred to as theultimate regime or the T2 regime, which is descibed as the least known and understood state of flow,with the main reason being the lack of experimental data.

We now consider the case of β = 0 for the cylinder. As we refine the mesh with respect to the errorin drag, we find that the Euler/G2 solution (or EG2 solution) approaches the potential solution, withseparation in one point (line) only, and with zero drag. But as the mesh is further refined we find thatthe potential solution is unstable, with the separation point generating vorticity and starting to oscillate.The oscillations are not simultaneous along the cylinder, instead there is a significant variation in thespanwise direction. The oscillating separation line generates streaks of vorticity in the streamwisedirection that undergo transition to turbulence downstream the cylinder, which is shown in Fig. 10-12,where an EG2 solution is plotted for a mesh with 75 000 mesh points. We note that as soon as thepotential solution looses stability, the drag increases and is sometimes even higher than for the case oflaminar separation at θ ≈ 90, see Fig. 13, where the time series of the drag for a EG2 solution on amesh with 150 000 nodes is shown.

The separation points generate vorticity, and sometimes structures similar to a regular von Karmanvortex street develops, with the fundamental difference that for the EG2 solution there is separationin one point only. Indeed, studying geophysical bluff body problems, such as the flow of air past theGuadalupe Island or the Canary Islands in Fig. 15, we find that these flows appear to share the featureof separation in one point only, which is consistent with the EG2 solution, se Fig. 14, rather thanin two points with a wake in between, which is the case of a standard von Karman vortex street atlow Reynolds numbers, see NASA, 2005. A circular cylinder is of course not a realistic model of thegeophysical flows in Fig. 15, which are no perfect cylinders but rather irregular 3d shapes closer to ahalf sphere or a surface mounted cube. Although, we believe that the computations indicate that forshapes of large dimensions, the separation may be well predicted using an EG2 model.

We note that this model is cheap since we do not have to resolve any boundary layers. The onlyparameter in the EG2 model is the discretization parameter h, and after some mesh refinement weexpect the EG2 solution to be independent of h with respect to certain mean value output, such as dragfor example. In particular, this means that we would be able to determine the dimensionless numbercD (up to a tolerance) using a computational model without any empirical constants.



Figure 10. EG2: Isosurfaces for vorticity: x1-component (left), x2-component (middle), and x3-component (right),at three different times (upper, middle, lower), in the x1x2-plane.





0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Figure 13. Time series of cD for an EG2 solution on a mesh with 150 000 nodes.



The importance of a reliable model for computing incompressible flow as Re→ ∞ will increase asthis regime is nowadays frequently reached in civil-, offshore and wind engineering as the size of thestructures increase.

Figure 14. EG2: x3-vorticity at two different times, in two different sections parallel to the x1x2-plane.

Figure 15. Clouds over the Guadalupe Islands (left) and the Canary Islands (right).

14.2. d’Alembert’s Paradox

EG2 offers a resolution of d’Alembert’s paradox, see Hoffman and Johnson, 2006c, which is differentfrom that suggested by Prandtl in 1904 (Prandtl, 1904) based on boundary layer effects from vanishingviscosity: EG2 shows that the zero-drag potential solution is unstable and instead a turbulent solutiondevelops with non-zero drag. Our solution differs from Prandtl’s in the following respects: (a) no



boundary layer before separation, (b) only one separation point in each section of the cylinder and (c)strong generation of vorticity in the streamline direction at the separation point.

14.3. Irreversibility in reversible systems

EG2 offers a resolution of the classical Loschmidt paradox (Loschmidt, 1876) asking how macroscopicirreversibility (as expressed by the Second Law of Thermodynamics) may occur in systems basedon reversible microscopic mechanics, see Hoffman and Johnson, 2006d. EG2 shows that the finiteprecision computation of G2 for the formally reversible Euler equations, satisfies the Second Law. EG2thus gives an explanation of the Second Law and irreversibility without using statistical mechanics asin the classical resolution suggested by Boltzmann (Boltzmann, 1964).

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