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Explicit Unsteady Navier–Stokes Solutions and Their Analysis Via Local Vortex Criteria

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Explicit Unsteady Navier–Stokes Solutions and Their Analysis Via Local Vortex Criteria Explicit unsteady Navier–Stokes solutions and their analysis via local vortex criteria Cite as: Phys. Fluids 32, 046603 (2020); https://doi.org/10.1063/5.0003245 Submitted: 31 January 2020 . Accepted: 06 April 2020 . Published Online: 22 April 2020 Tiemo Pedergnana , David Oettinger, Gabriel P. Langlois, and George Haller Phys. Fluids 32, 046603 (2020); https://doi.org/10.1063/5.0003245 32, 046603 © 2020 Author(s). Physics of Fluids ARTICLE scitation.org/journal/phf Explicit unsteady Navier–Stokes solutions and their analysis via local vortex criteria Cite as: Phys. Fluids 32, 046603 (2020); doi: 10.1063/5.0003245 Submitted: 31 January 2020 • Accepted: 6 April 2020 • Published Online: 22 April 2020 Tiemo Pedergnana,1,a) David Oettinger,2 Gabriel P. Langlois,3 and George Haller4,b) AFFILIATIONS 1 Institute of Energy Technology, ETH Zürich, Sonneggstrasse 3, 8092 Zürich, Switzerland 2ELCA Informatik AG, Steinstrasse 21, 8003 Zürich, Switzerland 3Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912, USA 4Institute for Mechanical Systems, ETH Zürich, Leonhardstrasse 21, 8092 Zürich, Switzerland a)Author to whom correspondence should be addressed: [email protected] b)Electronic mail: [email protected] ABSTRACT We construct a class of spatially polynomial velocity fields that are exact solutions of the planar unsteady Navier–Stokes equation. These solutions can be used as simple benchmarks for testing numerical methods or verifying the feasibility of flow-feature identification prin- ciples. We use examples from the constructed solution family to illustrate the deficiencies of streamline-based feature detection and those of the Okubo–Weiss criterion, which is the common two-dimensional version of the broadly used Q-, Δ-, λ2-, and λci-criteria for vortex-detection. Our planar polynomial solutions also extend directly to explicit, three-dimensional unsteady Navier–Stokes solutions with a symmetry. Published under license by AIP Publishing. https://doi.org/10.1063/5.0003245., s I. INTRODUCTION provided that A˙ (t)+A2(t) is symmetric. Note that all such solutions In this paper, we address the following question: For what are universal (i.e., independent of the Reynolds number) because 2 the viscous forces vanish identically on them. Reviews of exact time-dependent vectors akj(t) ∈ R does the two-dimensional (2D) velocity field, Navier–Stokes solutions tend to omit a discussion of the Craik– n m Criminale solutions although they list several specific spatially lin- u(x, t) = ∑ ∑ a (t)xkyj, (1) ear steady solutions for concrete physical settings (see the work of kj 4 5–7 8 j=0 k=0 Berker, Wang, and Drazin and Riley ). For solutions with at least solve the incompressible Navier–Stokes equation with the spatial quadratic spatial dependence, no general results of the specific form variable x = (x, y) ∈ 2 and the time variable t ∈ ? Answering this (1) have apparently been derived. R R 9 question enables one to produce a large class of exact Navier–Stokes In a related work, Perry and Chong outlined a recursive pro- solutions for numerical benchmarking and for verifying theoreti- cedure for determining local Taylor expansions of solutions of the cal results on simple, dynamically consistent, unsteady flow mod- Navier–Stokes equation up to any order when specific boundary 10 els. For linear velocity fields [akj(t) ≡ 0 for k, j > 1], general exis- conditions are available. Bewley and Protas showed that near tence conditions are detailed by Majda1 and Majda and Bertozzi.2 a straight boundary, all the resulting Taylor coefficients can be A more specific form of these spatially linear solutions is givenby expressed as functions and derivatives of the skin friction and the Craik and Criminale,3 who obtain that any differentiable function wall pressure. The objective in these studies is, however, the recur- sive construction of Taylor coefficients for the given boundary a00(t) and any differentiable, zero-trace matrix A(t) generate a linear Navier–Stokes solution u(x, t) in the form conditions rather than a derivation of the general form of exact poly- nomial Navier–Stokes velocity fields of finite order. We also men- 11 u(x, t) = a00(t) + A(t)x, (2) tion the work of Bajer and Moffatt, who construct exact, steady Phys. Fluids 32, 046603 (2020); doi: 10.1063/5.0003245 32, 046603-1 Published under license by AIP Publishing Physics of Fluids ARTICLE scitation.org/journal/phf three-dimensional (3D) Navier–Stokes flows with quadratic spatial of Eq. (3) is a universal solution of the planar Navier–Stokes equa- dependence for which the flux of the velocity field through the unit tion if and only if sphere vanishes pointwise. The linear part of the velocity field (1) can already have arbi- Δu ≡ 0, (5) trary temporal complexity, but remains spatially homogeneous by construction. The linear solution family (2) identified by Craik and i.e., if it is a harmonic solution. Hence, when looking for univer- Criminale3 cannot, therefore, produce bounded coherent flow struc- sal solutions of the form (1) that satisfy (4), we look for solutions tures. As a consequence, these linear solutions cannot yield Navier– u(x, t) whose components are harmonic polynomials in x with time- Stokes flows with finite coherent vortices or bounded chaotic mixing dependent coefficients. Since the viscous terms in the Navier–Stokes zones. equation vanish for harmonic flows, the universal solutions we find Higher-order polynomial vector fields of the form (1), how- will also be the solutions of Euler’s equation. Our main result is as ever, are free from these limitations, providing an endless source follows: of unsteady and dynamically consistent examples of flow structures away from boundaries. We will construct specific examples of such Theorem 1. An nth-order, unsteady polynomial velocity field flows and illustrate how the instantaneous streamlines, as wellas u(x, t) = (u(x, y, t), v(x, y, t)) of the spatial variable x = (x, y) the 2D version of the broadly used Q-criterion of Hunt, Wray, is a universal solution of the planar, incompressible Navier–Stokes 12 and Moin, the Δ-criterion in Ref. 13, the λ2-criterion in Ref. 14, equation (3) if and only if and the λci-criterion (or swirling-strength criterion) in Ref. 15, fail 1 −y n a (t) b (t) Re(x + iy)k in describing the fluid particle behavior correctly in these exam- u(x, t) = h(t) + ω( ) + ∑( k k )( ) (6) x b (t) −a (t) k ples. We will also point out how the two-dimensional exact solu- 2 k=1 k k Im(x + iy) tions we construct explicitly extend to unsteady 3D Navier–Stokes 2 solutions. holds for some arbitrary smooth functions h : R → R , ak, bk∶ R → R and an arbitrary constant ω ∈ R, where ω coincides with the constant scalar vorticity field of u(x, t). II. UNIVERSAL NAVIER–STOKES SOLUTIONS We rewrite the incompressible Navier–Stokes equation (under Proof: To prove that a universal solution u(x, t) must be pre- potential body forces) for a velocity field u(x, t) in the following cisely of the form given in (6), we recall from the work of Andrews, form: Askey, and Roy16 that a basis of the space of kth-order, homogeneous harmonic polynomials of two variables is given by @u p + V + (u ⋅ ∇)u − νΔu = ∇[− ], (3) @t ρ {Re[fk(x, y)], Im[fk(x, y)]}, k (7) fk(x, y) ∶= (x + iy) , k = 0, 1, 2, ::: . where ν ≥ 0 is the kinematic viscosity, ρ > 0 is the density, p(x, t) is the pressure field, and V(x, t) is the potential of external body Hence, the most general form of a polynomial u(x, t) satisfying the forces, such as gravity. We note that the left-hand side of (3) is a gra- universality condition (5) is dient (i.e., conservative) vector field. By classic results in potential theory, a vector field is conservative on a simply connected domain n α (t) β (t) Re[ f (x, y)] u(x, t) = ∑( k k )( k ) (8) γ (t) δ (t) Im[ f (x, y)] if and only if its curl is zero. For a 2D vector field, this zero-curl k=0 k k k condition is equivalent to the requirement that the Jacobian of the vector field is zero, as already noted in the construction of linear for some smooth scalar-valued functions αk(t), βk(t), γk(t), and 3 Navier–Stokes solutions by Craik and Criminale. On open, simply δk(t). By the Cauchy–Riemann equations for holomorphic complex connected domains, therefore, a sufficient and necessary condition functions, f k(x, y) must then satisfy for u(x, t) to be a Navier–Stokes solution is given by @Ref @Imf k = k , (9) @u @u T @x @y ∇[ + (u ⋅ ∇)u − νΔu] = (∇[ + (u ⋅ ∇)u − νΔu]) , (4) @t @t @Ref @Imf k = − k . (10) which no longer depends on the pressure and the external body @y @x force potential V. Substituting (1) into (4) and equating equal pow- ers of x and y in the off-diagonal elements of the matrices on the Under these conditions, requiring the divergence of u(x, t) in (8) to opposite sides of the resulting equation, we obtain conditions on vanish is equivalent to the unknown coefficients of the spatially polynomial velocity field u(x, t). n @Ref @Imf @Ref @Imf ( k k k k ) By a universal solution of Eq. (3), we mean a solution u(x, t) ∑ αk + βk + γk + δk = @x @x @y @y on which viscous forces identically vanish, rendering the pressure k 0 n p(x, t) independent of the Reynolds number. Note that all spatially @Refk @Refk = ∑((αk + δk) + (γk − βk) ) ≡ 0.
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