The Encyclopedia of Computational Mechanics chapter 'Computability and Adaptivity in CFD' (Hoffman and Johnson, 2004) identifies the principal open problem of computational fluid dynamics: whether mean-value outputs of very-high-Reynolds-number turbulent flows, such as time-averaged drag and lift, can be computed with guaranteed, rigorous error control. Within the chapter's General Galerkin (G2) framework, output accuracy is governed by stability factors of a linearized dual (adjoint) problem, which are observed numerically to be of moderate size for mean-value outputs, apparently due to cancellation within oscillatory reaction coefficients that no theoretical estimate captures. The subsequent record — the authors' 2017 second edition of the chapter, a computational study of dual-problem stability (2012), a parameter-free adaptive method (2015), a conjectural energy-stability analysis (2021), and the adjacent adjoint-based error-estimation literature — shows that the empirical few-percent computability of turbulent outputs stands, but the rigorous theory connecting turbulence statistics to dual stability, and with it any guarantee of output accuracy for high-Reynolds-number turbulent flow, does not exist. The problem remains open.

Background

Most flows of engineering interest — around cars, aircraft, and ships — are turbulent: at high Reynolds number ReRe (the ratio of inertial to viscous forces in the Navier–Stokes equations of fluid motion) the flow becomes chaotic, with vortical structures spanning an enormous range of scales. Direct numerical simulation (DNS), which resolves all scales down to the smallest dissipative ones, needs a number of degrees of freedom that grows steeply with ReRe and is infeasible in practice; what practitioners instead ask for are mean-value outputs, such as the drag or lift force on a body averaged over time, rather than pointwise velocity and pressure fields. The open question is whether such outputs can ever be computed with a guaranteed error bound.

Hoffman and Johnson's 2004 Encyclopedia of Computational Mechanics chapter 'Computability and Adaptivity in CFD' gave this question a precise computational formulation. Their General Galerkin (G2) method is an adaptive finite element discretization of the (incompressible) Navier–Stokes or Euler equations in which a weighted least-squares term based on the residual (the amount by which the computed solution fails to satisfy the discrete equations) supplies numerical dissipation that serves in place of an explicit turbulence model; the scalar ϵ\epsilon measures the least-squares residual and is interpreted as the modeled turbulent dissipation. For a chosen output M(u)M(u) (say, mean drag), the output error of the computed solution UU is bounded by a duality-based a posteriori estimate: the residual of UU weighted by the solution of a dual (adjoint) problem — the Navier–Stokes equations linearized at UU and integrated backward in time with the output functional as data. The norms of these weights, the stability factors, measure the sensitivity of the output to numerical errors.

The empirical findings of the chapter and the companion papers are striking: for pointwise outputs the dual weights grow exponentially, so pointwise values of turbulent flow are not computable, while for mean-value outputs such as time-averaged drag and lift around bluff bodies (non-streamlined bodies, such as a circular cylinder, behind which the flow separates) the computed dual problems are effectively stable and the stability factors are of moderate size, allowing a few-percent tolerance with roughly 10510^5 mesh points for simple geometries (about 10610^6 for complex ones, per the 2017 second edition). This is the basis of what the authors call weak uniqueness by computation: mean outputs are insensitive to perturbations of mesh and model even though pointwise values are not. The difficulty is theoretical: the dual problem contains rapidly oscillating reaction coefficients, and the observed moderate stability appears to arise from cancellation effects for which no analysis accounts — the source records this as seeming impossible to capture in purely theoretical estimates. On the rigorous side, the framework's existence and error results concern ϵ\epsilon-weak solutions (approximate weak solutions whose equation and continuity residuals are bounded by ϵ\epsilon in a least-squares sense) and the ϵ\epsilon–hh error relations, which rely on auxiliary assumptions such as boundedness of the discrete maximum velocity max⁡∣U∣\max|U|; near walls, G2 uses a slip-with-friction boundary condition whose friction coefficient β\beta must be calibrated to reproduce physical skin friction rather than being determined from first principles. The chapter also studies EG2, G2 applied to the Euler equations (zero viscosity), whose computations reproduce turbulence-like separation and vortex shedding, raising the question of whether this captures the correct Re→∞Re\to\infty limiting behavior.

Problem Statement

Can mean-value outputs of incompressible turbulent flow at very high Reynolds number — such as the time-averaged drag or lift on a body — be computed with guaranteed, rigorous error control, rather than the currently available empirical demonstration? In the G2/duality framework of Hoffman and Johnson, where the question is precisely posed, this has a concrete open theoretical core: determine whether such a rigorous guarantee can exist — either by establishing a theory connecting the statistical structure of high-Reynolds-number turbulent flow to the stability of the linearized dual (adjoint) problem, or by proving that no such guarantee is attainable in the demonstrated setting. Such a theory should explain and bound the numerically observed moderate size of dual stability factors for mean-value outputs — currently attributed only to cancellation within oscillatory reaction coefficients and lacking any theoretical account — so that the demonstrated few-percent computability of such outputs becomes a guaranteed a posteriori error bound of the form ∣M(u^)−M(U)∣≤h R(U) S0(ψ^)≤TOL|M(\hat u) - M(U)| \le h\,R(U)\,S_0(\hat\psi) \le TOL, where R(U)R(U) is the residual of the computed solution and S0(ψ^)S_0(\hat\psi) is the stability factor of the dual problem. The guarantee must hold at least for the setting in which empirical computability was demonstrated in the source — three-dimensional incompressible turbulent flow at high Reynolds number around bluff bodies, with mean-value outputs such as time-averaged drag and lift at a few-percent tolerance — while the overall target remains mean-value outputs of high-Reynolds-number turbulent incompressible flow as posed by the source chapter. Related open aspects of the same program discussed in the sources, though not jointly necessary conditions for this guarantee, include removal of auxiliary assumptions such as boundedness of the discrete maximum velocity max⁡∣U∣\max|U| in the ϵ\epsilon-weak-solution analysis, a first-principles wall treatment replacing the empirically calibrated wall-model friction coefficient β\beta, and the theoretical delineation of the flows, outputs, and boundary conditions for which dual problems are effectively stable.

References

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  2. J. Hoffman, J. Jansson, N. Jansson, R. V. de Abreu and C. Johnson, 'Computability and Adaptivity in CFD', Encyclopedia of Computational Mechanics, 2nd ed., Wiley, 2017, pp. 1-22, DOI: 10.1002/9781119176817.ecm2057, https://doi.org/10.1002/9781119176817.ecm2057
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