Dean Instability in Double-Curved Channels J.-D. Debus,∗ M. Mendoza,† and H. J. Herrmann‡ ETH Z¨urich, Computational Physics for Engineering Materials, Institute for Building Materials, Wolfgang-Pauli-Str. 27, HIT, CH-8093 Z¨urich (Switzerland) We study the Dean instability in curved channels using the lattice Boltzmann model for generalized metrics. For this purpose, we first improve and validate the method by measuring the critical Dean number at the transition from laminar to vortex flow for a streamwise curved rectangular channel, obtaining very good agreement with the literature values. Taking advantage of the easy implementation of arbitrary metrics within our model, we study the fluid flow through a double- curved channel, using ellipsoidal coordinates, and study the transition to vortex flow in dependence of the two perpendicular curvature radii of the channel. We observe not only transitions to 2- cell vortex flow, but also to 4-cell and even 6-cell vortex flow, and we find that the critical Dean number at the transition to 2-cell vortex flow exhibits a minimum when the two curvature radii are approximately equal. PACS numbers: 47.11.-j, 47.20.-k I. INTRODUCTION Numerical simulations of curved channel flow have also been popular, see for example Finlay et al. [1, 5, 6], who Curved channel flow has attracted fluid dynamics re- extended Dean’s stability criterion to channels of differ- searchers for over a hundred years because of its practi- ent aspect ratios η = ri/ro, using high-accuracy linear cal applications in engineering and fascinating features at stability analysis. high Reynolds numbers. At low Reynolds numbers, the All the studies mentioned above deal with a very ide- flow through a curved duct is laminar, and its velocity alized channel geometry, namely with channels having profile resembles the parabolic profile of plane channel a rectangular cross section, which are uniformly curved flow. Following Ref. [1], we will refer to this type of along the streamwise direction only. Thinking of practi- flow as “curved channel Poiseuille flow”. For increasing cal engineering applications (e.g. water ducts), however, Reynolds number, the flow pattern is determined by cen- channels do not possess a perfectly rectangular cross sec- trifugal forces until it becomes unstable and bifurcates to tion, but can also be bent perpendicularly to the stream- secondary flow. In 1928, W. R. Dean showed analytically wise direction for instance under the influence of external for the case of a narrow channel that the fully developed forces such as gravity. The resulting cross section be- flow between two concentric cylinders (see Fig. 1) can comes the section of a circular ring, such that the chan- be characterized by a single non-dimensional parameter, nel possesses a second, cross-sectional curvature. In this the Dean number e, which depends on the Reynolds paper, we study the Dean instability in such a double- number e as wellD as on the channel geometry [2]. It is curved channel, which, to the best of our knowledge, has defined asR never been done before. Introducing a second curvature, we expect qualitative deviations from the behaviour of d the Dean flow as compared to the idealized cylindrical e := e , D R · r case, since the translational symmetry along the cylinder R axis is then broken. where d = r r is the channel width, r , r are the o i i o For the double-curved channel, we study the flow at radii of the inner− and outer cylinder respectively and different Dean numbers and observe a bifurcation from denotes the curvature radius of the inner wall (for curved channel Poiseuille flow to vortex flow, which is aR cylinder, = r ). In particular, Dean performed a i similar to the vortex flow in the cylindrical channel, show- stability analysisR to show that flow between two concen- arXiv:1411.3133v1 [physics.flu-dyn] 12 Nov 2014 ing a pair of counter-rotating vortices. We measure the tric cylinders becomes unstable for small perturbations critical Dean number at the bifurcation point in depen- when the Dean number e exceeds a critical value of D dence of the two perpendicular curvatures. ec 36. Above this value, pairs of counter-rotating streamwise-orientedD ≈ vortices develop, which is known as First, we vary the streamwise curvature radius θ while keeping the cross-sectional curvature radius R Dean vortex flow. The theoretical predictions by Dean Rφ have been confirmed in various experiments, see for ex- fixed. Surprisingly, we find that the lowest Dean num- ample H¨ammerlin [3] or Brewster, Grosberg & Nissan [4]. ber at which the Dean instability occurs corresponds to the configuration in which both curvature radii are ap- proximately equal, θ φ. For channels with both weaker or stronger streamwiseR ≈ R curvature, we measure an ∗ [email protected] (almost linear) increase of the critical Dean number with † [email protected] . For strongly-curved channels, we even observe a sec- Rθ ‡ [email protected] ond bifurcation to vortex flow with 4 counter-rotating 2 vortices. II. LATTICE BOLTZMANN METHOD Secondly, we vary the cross-sectional curvature radius while keeping the streamwise curvature fixed. Again, In this section, we will shortly review the method used φ to simulate the motion of the fluid. For further details weR find a minimum of the critical Dean number for the case , consistent with the first study. For in- we refer to [12, 13]. θ φ The method is based on the Boltzmann equation, creasingR ≈ R, the curvature perpendicular to the stream- φ which describes the motion of fluid particles in terms of wise directionR becomes smaller and smaller, such that the a distribution function f. In a curved three-dimensional double-curved channel approaches a cylindrical geome- Riemann space, the Boltzmann equation reads try in the limit . Correspondingly, for increas- Rφ → ∞ ing φ, the critical Dean number for the double-curved ∂f ∂f ∂f R + ξi + F i = [f] (1) channel approaches the value of the critical Dean number ∂t ∂xi ∂ξi C for the cylindrical channel in our simulations. [12], where f = f(xi,pi) denotes the distribution function, which depends on the space coordinates xi = (x1, x2, x3) as well as on the momentum pi = For the simulations, we use the lattice Boltzmann m(ξ1, ξ2, ξ3), where m and ξi denote the mass and ve- (LB) method, which has been developed during the last locity of the fluid particles respectively, and the mass is decades to simulate fluids by means of simple arithmetic set to m = 1. We are using the Einstein sum convention operations instead of discretizing and solving the com- throughout the whole paper, i.e. Latin indices run over plicated macroscopic equations of continuum fluid me- the spatial directions 1 to 3. The spatial metric enters the chanics. The method itself is based on the Boltzmann Boltzmann equation through the force F i := Γi ξj ξk, kinetic equation, which describes the motion of the mi- − jk which depends on the Christoffel symbols Γi and thus croscopic fluid particles instead of the macrosopic con- jk drives the particles along the geodesics of the curved tinuum. Because of its simplicity and straightforward space. Collisions between fluid particles are accounted parallelizability, the LB method has gained more and for by the collision operator [f], for which we use the more popularity among scientists and engineers in the Bhatnagar-Gross-Krook (BGK)C approximation [15] past. A review about the LB method is given in Ref. [7]. While it was originally designed to solve fluid flows, f f eq [f]= − , the LB method has even been applied to electrodynamics C − τ [8] and magnetohydrodynamics [9] as well as relativistic eq [10] and ultra-relativistic flows [11]. Most of the LB ap- where τ denotes the relaxation parameter and f is the plications use standard Cartesian coordinates (e.g. for Maxwell-Boltzmann equilibrium distribution. The latter fluid flow in rectangular cavities), which cannot be ap- is given by plied straightforwardly to more complex curved geome- eq ρ 1 i i j j tries. However, the lattice Boltzmann method has re- f = exp ξ u gij ξ u 3/2 −2θ − − cently been extended to general metrics being defined by (2πθT ) T a metric tensor [12, 13], which offers a variety of inter- i [12], where θT is the normalized temperature, ρ and u esting new applications. With this extension at hand, it denote the macroscopic density and velocity of the fluid becomes possible to simulate fluids in arbitrary geome- (normalized by the speed of sound cs) and gij are the tries, while standard LB methods are restricted to simple components of the metric tensor g. In the following, we geometries. will always assume the isothermal case θT = 1. In this paper, we improve the method of Ref. [12] by The macroscopic density ρ and velocity ~u are given increasing the accuracy of the forcing term in the LB by the zeroth and first order moment of the distribution equation. The improvement is validated for the case of a function, cylindrical channel, for which we measure the dependence ρ = f √g d3ξ, ρui = f ξi√g d3ξ, (2) of the critical Dean number on the channel aspect ratio Z Z η = ri/ro. Comparing our results to the numerical values given in Ref. [6] by Finlay et al., we find very good where √g d3ξ := √det g dξ1dξ2dξ3 denotes the invariant agreement. volume element of the momentum space. Since conser- vation of mass and momentum are intrinsic features of Since our model can handle arbitrary geometries, we the Boltzmann equation, the macroscopic density ρ and can easily introduce a second perpendicular curvature in velocity ui automatically fulfil the hydrodynamic conser- the channel by choosing ellipsoidal coordinates.