Dean Instability in Double-Curved Channels

arXiv:1411.3133v1 [physics.flu-dyn] 12 Nov 2014 where hnteDa number perturbations D small Dean for the concen- unstable when two becomes between flow cylinders that tric show and to analysis respectively stability cylinder cylinder, outer a and inner R the of radii aebe ofimdi aiu xeiet,sefrex- [4]. for Nissan see & Grosberg Dean experiments, Brewster, by various or Hämmerlin [3] ample in predictions confirmed theoretical been The have as flow. known vortex is Dean which develop, vortices streamwise-oriented endas defined number o ewe w ocnrcclnes(e i.1 can number 1) parameter, non-dimensional Dean single Fig. the a (see developed by fully cylinders characterized the concentric be that two channel narrow between a flow analytically of showed case Dean R. the W. for to 1928, bifurcates In of and flow. unstable type secondary becomes cen- this it by until determined to increasing forces is For trifugal refer pattern flow will flow”. the number, Poiseuille we Reynolds channel channel [1], “curved plane as velocity Ref. flow of its Following profile and parabolic laminar, flow. is the duct resembles the curved numbers, profile Reynolds a low through At flow practi- numbers. at its Reynolds features fascinating of high and because engineering years in applications hundred cal a over for searchers ∗ ‡ † e [email protected] [email protected] [email protected] uvdcanlflwhsatatdfli yaisre- dynamics fluid attracted has flow channel Curved c eoe h uvtr aiso h ne al(for wall inner the of radius curvature the denotes ≈ d 6 bv hsvle ar fcounter-rotating of pairs value, this Above 36. R = e R r swl so h hne emty[] tis It [2]. geometry channel the on as well as ASnmes 71.j 47.20.-k 47.11.-j, numbers: PACS flow mini vortex a o exhibits 6-cell We flow even vortex and 2-cell equal. channel. to 4-cell approximately the transition to the of also at but number radii flow, curvature vortex perpendicular cell t two st study we the and a model, coordinates, of for ellipsoidal our using flow literature m within channel, vortex the metrics curved the with to arbitrary validate agreement of laminar and good implementation from improve very transition first obtaining the channel, we at purpose, number this Dean For metrics. o nttt o uligMtras ofagPuiSr 27 Wolfgang-Pauli-Str. Materials, Building for Institute esuyteDa ntblt ncre hnesuigtela the using channels curved in instability Dean the study We − = .INTRODUCTION I. r r i i D .I atclr enpromda performed Dean particular, In ). D stecanlwidth, channel the is e e hc eed nteReynolds the on depends which , := D R e T ¨rc,CmuainlPyisfrEgneigMateri Engineering for Physics Computational Zürich, ETH enIsaiiyi obeCre Channels Double-Curved in Instability Dean e xed rtclvleof value critical a exceeds · r .D Debus, J.-D. R d , r i ∗ r , o .Mendoza, M. r the are enpplr e o xml Finlay also example have for flow see channel popular, curved been of simulations Numerical xeddDa’ tblt rtro ocanl fdiffer- of ratios channels aspect to ent criterion stability Dean’s extended h ofiuaini hc ohcrauerdiaeap- are radii curvature to both equal, corresponds which proximately num- occurs in Dean instability configuration lowest Dean the the the which that at radius find ber we curvature Surprisingly, cross-sectional fixed. the keeping while rtclDa ubra h iucto on ndepen- curvatures. in perpendicular point two the bifurcation the measure of the We dence at number vortices. Dean counter-rotating is critical of show- which pair channel, cylindrical a flow, the ing in vortex flow vortex to from the to flow bifurcation similar Poiseuille a observe channel and curved numbers Dean different cylindrical broken. cylinder idealized then the is the along axis symmetry to translational of compared the behaviour since as case, the flow from Dean deviations curvature, the second qualitative a expect has double- Introducing knowledge, we a before. our of done such best been in never the instability to which, Dean channel, this the chan- curved In study the curvature. that we be- cross-sectional such paper, section second, ring, a cross circular possesses a resulting nel of The section the gravity. comes external as of such influence stream- the the forces under to instance perpendicularly for bent direction sec- be wise cross also rectangular can but perfectly a tion, however, possess ducts), not water do practi- channels (e.g. of applications Thinking curved engineering only. uniformly cal direction are streamwise having which the section, channels along cross with rectangular namely a geometry, channel alized analysis. stability n iucto ovre o ih4counter-rotating 4 with flow vortex to bifurcation ond ams ier nraeo h rtclDa ubrwith number Dean critical the R an of measure increase we linear) curvature, (almost streamwise stronger or weaker † θ n .J Herrmann J. H. and is,w aytesraws uvtr radius curvature streamwise the vary we First, at flow the study we channel, double-curved the For ide- very a with deal above mentioned studies the All o togycre hnes eee bev sec- a observe even we channels, strongly-curved For . etasto ovre o ndependence in flow vortex to transition he I,C-03Zürich (Switzerland) CH-8093 HIT, , u hntetocrauerdiare radii curvature two the when mum aus aigavnaeo h easy the of advantage Taking values. d h udflwtruhadouble- a through flow fluid the udy generalized for model Boltzmann ttice n efidta h rtclDean critical the that find we and , srentol rniin o2- to transitions only not bserve to ymauigtecritical the measuring by ethod temiecre rectangular curved streamwise η R = θ r ‡ R ≈ i /r als, o sn ihacrc linear high-accuracy using , φ o hneswt both with channels For . tal. et 1 ,6,who 6], 5, [1, R R φ θ 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 conservation 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. To this vation equations, which can be shown rigorously by a end, we only have to adapt the metric tensor, which is Chapman-Enskog expansion [16]. In covariant form, the much simpler then deriving a new expression for the LB macroscopic conservation equations read equation for each special choice of coordinates, as it is ∂ρ ∂ commonly done in the literature for very simple geome- + ρui =0, ρui + T ij =0, tries (see e.g. Ref. [14]). ∂t ∇i ∂t ∇j 3

λ ~cλ wλ tion (1) becomes

2 1 (0, 0, 0) 2025 5045 1507√10 i i i ∆t eq − fλ(x + c ∆t,t + ∆t) fλ(x ,t)= (fλ f ) + ∆t λ, λ − − τ − λ F 2,3 ( 1, 0, 0) (3) ± 37 91 4,5 (0, 1, 0) 5 10 40 ± √ − where the left-hand side represents free streaming, 6,7 (0, 0, 1) ± whereas the right-hand side corresponds to particle collisions. The forcing term contains the information 8-11 ( 1, 1, 0) Fλ ± ± 1 about the geometry of the space, encoded in a combi- 12-15 ( 1, 0, 1) 55 17√10 ± ± 50 − nation of Christoﬀel symbols. The relaxation parameter 16-19 (0, 1, 1) ± ± τ is directly related to the dynamic shear viscosity µ as 1 well as to the kinematic viscosity ν by 20-27 ( 1, 1, 1) 1600 233√10 730 ± ± ± − 1 28,29 ( 3, 0, 0) µ = ρν = ρ τ c2∆t. (4) ± 1 s 30,31 (0, 3, 0) 295 92√10 − 2 ± 16200 − 32,33 (0, 0, 3) In order to obtain discrete expressions for the equilib- ± eq 1 rium distribution f and for the forcing term λ, we ex- 34-41 ( 3, 3, 3) 129600 130 41√10 λ ± ± ± − pand the distribution function into tensor HermiteF poly- nomials, deﬁned by TABLE I. Discrete velocity vectors cλ of the D3Q41 lattice and the corresponding weights wλ in the Hermite expansion. i1,...,in ~ n ~ −1 ∂ ∂ ~ (n) (ξ) = ( 1) w(ξ) w(ξ), H − ∂ξi1 ··· ∂ξin where denotes the covariant derivative (Levi-Civita where w(ξ~), the weight function, is given by connection)∇ and T ij is the energy-stress tensor. Explic- itly, the energy-stress tensor is given by T ij = Pgij + 1 1 2 i j lj i il j w(ξ~)= exp ξ~ . ρu u µ(g lu + g lu ), where P = ρθ is the hy- (2π)3/2 −2| | drostatic− pressure,∇ µ is∇ the dynamic shear viscosity and gij denote the components of the inverse metric tensor. This yields

∞ In order to obtain the lattice version of the Boltz- 1 f(~x, ξ,t~ )= w(ξ~) ai1,...,in (~x, t) i1,...,in (ξ~), (5) mann equation, the coordinate and momentum space n! (n) H(n) are discretized on a (sufficiently symmetric) lattice, and nX=0 all vectors are expressed in terms of a commuting ba- where a are the coefficients of the expansion. For the ∂ ∂ ∂ (n) e e e eq sis ( 1, 2, 3) = ( ∂x1 , ∂x2 , ∂x3 ), which defines the co- equilibrium distribution, f = f , the first four expansion ordinate frame on the lattice. The minimum configura- coefficients are given by tion of discrete velocities to fulfil all necessary symmetry relations is given by the D3Q41 lattice proposed in eq ′ eq,i ′ i eq,ij ′ 2 ij ′ i j a = ρ , a = ρ u , a = ρ cs∆ + ρ u u , Ref. [17]. This lattice contains a set of 41 discrete ve- (0) (1) (2) 41 eq,ijk ′ 2 ij k jk i ki j ′ i j k locities ~c (see Table I), which are normalized by a = ρ cs ∆ u + ∆ u + ∆ u + ρ u u u , (6) { λ}λ=1 (3) the speed of sound cs, i.e. ξ~λ = ~cλ/cs. The value of the speed of sound for this specific lattice is given by where ∆ij := gij δij is a measure for the deviation 2 from flat space and− factors c2 account for a normaliza- cs = 1 2/5. From the discretized distribution func- s − ~ p ~ tion of the velocities ξ ~cλ/cs. Now, we obtain the tion fλ(~x, t) := f(~x, ξλ,t), the macroscopic quantities of → the fluid, i.e. the density ρ and the macroscopic velocity lattice version of the equilibrium distribution by apply- ~u, are recovered by taking moments of the distribution ing the Gauss-Hermite quadrature rule, which in this case ~ function, is equivalent to replacing w(ξ) by wλ, where the discrete weights wλ are given in Table I. For the D3Q41 lattice, 41 41 Gauss-Hermite quadrature preserves the first four mo- ′ i ′ i ments of the distribution function exactly. Therewith, fλ = ρ , cλfλ = ρ u , λX=1 λX=1 the lattice equilibrium function is given by

′ eq eq 1 eq,i i 1 eq,ij i j 2 ij where ρ = ρ/√g (the factor √g in the invariant integra- fλ = wλ a(0) + 2 a(1) cλ + 4 a(2) cλcλ csδ tion measure in Eq. (2) is absorbed into a redeﬁnition of cs 2!cs − 1 the ﬂuid density). + aeq,ijk ci cj ck c2 δij ck + δjkci + δkicj . 3!c6 (3) λ λ λ − s λ λ λ In terms of discretized quantities, the Boltzmann equa- s 4

In order to calculate the forcing term , we rewrite Fλ ∞ ∂f 1 F i = w ai1,...,in F i i,i1,...,in , (7) ∂ξi − n! (n) H(n+1) nX=0 where we have expressed f in terms of its Hermite expansion (5). As an improvement of the forcing term in Ref. [13], we do not approximate all the expansion coeﬃcients a(n) by the coeﬃcients of the equilibrium distribution (6), but set

a = aeq , ai = aeq,i, aij = aeq,ij σij , (0) (0) (1) (1) (2) (2) − ij 1 i j eq where σ = (1 2τ ) λ cλcλ (fλ fλ ) denotes the − − 1 − stress tensor, and the factorP (1 2τ ) accounts for discrete lattice effects. Finally, we obtain− the discrete forcing term by plugging these coefficients into Eq. (7) (using normalized velocities ~cλ), and truncating the expansion at third order: FIG. 1. (Color online) Geometry of the curved channel in 1 1 i i i j i j 2 ij cylindrical coordinates (r,θ,z). The colored cross-sections de- λ = wλ 2 a(0)Fλcλ + 4 a(1)Fλ cλcλ csδ F cs cs − pict the vorticity of the axisymmetric flow, where the blue (lower) and red (upper) spots correspond to low and high 1 ij k i j k 2 ij k jk i ki j + a F c c c c δ c + δ c + δ c . values of the vorticity, respectively. The dashed lines indicate 2c6 (2) λ λ λ λ − s λ λ λ s the periodicity of the channel in the θ- and z-direction. where F i = Γi cj ck . Additional external forces can be λ − jk λ λ added straightforwardly by replacing F i F i + F i . λ → λ ext Depending on the aspect ratio of the channel as well Having all ingredients at hand for the LB equation (3), as on the Reynolds number e, the primary Poiseuille the LB algorithm can be applied as usual: After assigning flow is expected to undergo a transitionR to secondary flow initial conditions to the macroscopic quantities ρ and ~u, (Dean flow) at a specific critical value of the Dean num- the distribution function f is successively updated time ber, step by time step according to the LB equation. This is done by dividing each time step into a free streaming d vθ d d step (corresponding to the left-hand side of Eq. (3)) and e := e = h i , (8) D R · r ν · r a collision step (corresponding to the right-hand side of R R Eq. (3)). Special care has to be taken for the boundary where vθ denotes the mean azimuthal fluid velocity, conditions, which will be addressed in the next sections ν = (τ h 1i/2) c2∆t is the kinematic viscosity, d = r r − s o − i in the context of concrete examples. is the channel width, ri and ro denote the inner and outer radius, respectively, and is the curvature radius of the R inner wall (for a cylinder = ri). The secondary flow III. VALIDATION: FLOW THROUGH occurs due to centrifugalR instabilities and is character- CYLINDRICAL CHANNEL ized by a pair of counter-rotating vortex tubes which are oriented along the stream direction. Using the LB For the validation of the improved model, we consider method as described in section II, we have modeled the the idealized case of flow in a closed cylindrical channel, axisymmetric fluid flow for different channel aspect ratios as depicted in Fig. 1. In an experimental application, η = ri/ro. In order to avoid staircase approximations at the channel would not be completely closed, but would the channel boundaries, we use cylindrical coordinates have a finite opening angle < 2π with open boundaries (x1, x2, x3) = (r,θ,z), which are perfectly adapted to the at the inlet and outlet. However, for long and narrow geometry of the channel. In these coordinates, the metric channels, as considered in this study, the ratio between tensor is given by the streamwise extent and the spanwise extent d (being relevant for the instability) is very small ( 10−2) such 1 0 0 ∼ 2 that possible finite-size effects at the inlet/outlet can be g = 0 r 0 , neglected. In order to drive the fluid, a pressure gradient 0 0 1 between the inlet and outlet of a channel can be applied.   In our simulations, we use an external force in the az- and we express all vector fields and tensors in terms of e e e ∂ ∂ ∂ imuthal direction instead, which for incompressible flow the standard basis ( r, θ, z) = ( ∂r , ∂θ , ∂z ), for example r θ z is equivalent to a pressure gradient. u = u er + u eθ + u ez. Since this basis is commuting, 5

Christoffel symbols can be calculated by in this study, both quantities are equivalent, since the vorticity is fully characterized by its streamwise compo- 1 ∂g ∂g ∂g θ Γi = gim mj + mk jk , nent, ω ). In particular, we are interested in the mean jk 2 ∂xk ∂xj − ∂xm absolute vorticity, averaged over the whole cross-section of the channel, which yields the only non-vanishing Christoffel symbols Γr = r, Γθ =Γθ =1/r. 1 ∂ur ∂uz θθ − rθ θr ωθ = r drdz, Since the cylindrical channel is translational invariant h i S Z ∂z − ∂r S in θ, we impose periodic boundaries in θ, which effec- tively reduces the three-dimensional problem to a two- where S = drdz denotes the cross-sectional area. At dimensional one. Since we are only interested in “two- the critical DeanR number ec, the vorticity is expected dimensional” vortex solutions (as they are called in Ref. to increase considerably. Indeed,D this behaviour has been [18]), each velocity component depends only on the span- observed in the simulations. wise directions r and z. For the simulations, we use a Exemplarily, Fig. 2 shows the average vorticity ωθ h i rectangular lattice of size Lr Lθ Lz = 128 1 256, depending on the Dean number e for a channel with as- where each lattice node is labeled× × by integer× lattice× in- pect ratio η = 0.80. The curveD agrees with the expecta- dices (ˆr, θ,ˆ zˆ) [1,Lr] [1,Lθ] [1,Lz]. The transfor- tions, showing a transition from zero-vorticity Poiseuille mation between∈ lattice× units and× physical units is given flow for e < 42 to vortex flow for e > 42. The be- D D by havior of the vorticity at ec 42 is indicative of an imperfect bifurcation, as canD be≈ seen in the recent work ˆ r = ri +ˆr ∆r, θ = θ ∆θ, z =z ˆ ∆z, by Haines et. al in Ref. [19]. At ec, two counter- · · · rotating vortex tubes form, which increaseD in strength where ∆r, ∆θ and ∆z denote the lattice spacings in the for higher Dean numbers. The colored pictures in Fig. radial, azimuthal and axial direction respectively. In our 2 illustrate the velocity streamlines on a cross-section of simulations, we set ∆r = ∆θ = ∆z = ∆t and ri = 1. the channel perpendicular to the stream direction at dif- Defining the aspect ratio as η = ri/ro, we obtain for the ferent Dean numbers. The colors represent the strength lattice spacing ∆r = (1 η)/(ηL ). In order to compare θ − r of the streamwise vorticity ω , where blue and red col- our results to the work by Finlay et al. [6], we only vary ors correspond to clockwise and counterclockwise rotat- the radial aspect ratio η = ri/ro, keeping the spanwise ing vortices, respectively. Fig. 2 also reveals a second aspect ratio fixed, Lz/Lr = 2. At the two channel walls bifurcation, which is indicated by a further increase of at r = r and r = r , we impose Dirichlet boundary con- i o the vorticity at ec2 54, where a second pair of vortex ditions on the fluid velocity, u(ro) = u(ri) = 0. This tubes begins toD form.≈ Fig. 3 shows the radial, stream- condition enters the algorithm through the equilibrium wise and axial velocity profiles versus the z-position for eq distribution f , which is evaluated with u =0 at r = ri the 4-cell vortex flow for η =0.80 and e = 56. and r = ro at each time step. In the θ- and z-directions, We have measured the critical DeanD number at the we use periodic boundary conditions. We note that by bifurcation from Poiseuille flow to 2-cell vortex flow for considering periodicity in z, a symmetry condition is im- different channel aspect ratios η = ri/ro. The results are posed on the solution, since the spanwise wave length of shown in Fig. 4, which depicts the dependence of the the Dean vortices is restricted to the values λ = 2Lz/k critical Dean number on the aspect ratio of the channel. (where k denotes the number of vortex cells). However, The errorbars result from the uncertainty in determin- the choice Lz/Lr = 2 is motivated by the fact that the ing the critical Dean number from the vorticity curve, channel cross-section is perfectly adapted to the 2-cell since the vorticity increases smoothly at the bifurcation vortex solution, since in this case the Dean cells can oc- point (as can be seen in Fig. 2). We have also compared cupy the total section. the present improved version of our method with the old The relaxation parameter τ can be used to tune the version used in a previous publication [12], where the mo- Reynolds number (and thus the Dean number) to the ments of the distribution function f in the forcing term parameter range of interest, where the bifurcation occurs. have been approximated by the moments of the equilib- For the simulations, we choose a τ = 0.9, which also rium distribution function f eq. Comparing both methods enhances fast convergence to the stationary state. to the numerical results by Finlay et al. [6] in Fig. 4, we In order to study the bifurcation from curved chan- observe that the present improved method agrees very nel Poiseuille flow to secondary Dean flow, we vary the well with the literature, whereas our old method leads Dean number e by varying the strength of the driving to deviations from the literature values. It can be seen force. At theD critical Dean number e , we expect the D c that the deviations between the old method and the im- formation of counter-rotating vortex tubes, which should proved method vanish for η 1. This means that the increase in strength for higher values of e. This can be → D error in the approximation of the forcing term in the old measured by calculating the vorticity of the flow, given method becomes negligible when the resolution is suffi- by ~ω = ~ ~u. (Note that some authors use the helic- ciently high. ∇× ity, ~h = ~u ~ω, instead, to analyze the Dean vortices. We have also plotted the second critical Dean number However, for× the streamwise oriented vortices considered e , at which the second bifurcation from 2-cell vortex D c2 6

60 old method improved method 55 Finlay et al. 50 c e

D 45

0.8 0.85 0.9 0.95 1 η = ri/ro

FIG. 4. (Color online) Critical Dean number ec at the bi- D θ furcation from Poiseuille flow to 2-cell vortex flow as function FIG. 2. (Color online) Average vorticity ω depending on of the aspect ratio η = ri/ro for a cylindrical channel. The the Dean number e for a cylindrical channelh i with aspect D results obtained by our improved method agree very well with ratio η = ri/ro = 0.80. As can be seen, the bifurcation from the literature values of Finlay et al. Poiseuille flow to 2-cell vortex flow occurs at Dec 42, fol- ≈ lowed by a second bifurcation to 4 vortices at ec2 54. The colored pictures depict the velocity streamlinesD on≈ a channel cross-section perpendicular to the stream direction. The colors represent the strength of the vorticity, where blue and red colors correspond to clockwise and counterclockwise rotating 56 vortices, respectively. 54

52 2 c

radial velocity ur e 50 0.010 D 0.005 48 0.000 −0.005 46 −0.010 44 0.8 0.85 0.9 0.95 1 streamwise velocity ut η = ri/ro 0.170 FIG. 5. (Color online) Second critical Dean number ec2 at 0.165 D 0.160 the bifurcation from 2-cell vortex ﬂow to 4-cell vortex ﬂow 0.155 as function of the aspect ratio η = ri/ro for a cylindrical channel. axial velocity uz 0.005

0.000

−0.005 Dependence on the Resolution

0 0.2 0.4 0.6 0.8 1 z/Lz To obtain an estimation of the resolution error in our FIG. 3. (Color online) Radial, streamwise and axial velocity simulations, we have measured the relative error of the profiles for the 4-cell vortex flow for a cylindrical channel with critical Dean number for different resolutions. The rela- aspect ratio η = 0.80 at Dean number e = 56 tive error is defined as the relative deviation of the criti- D cal Dean number from the corresponding reference value given by Finlay et al. in Ref. [6]. Fig. 6 depicts the dependence of the relative error on the grid resolution for flow to 4-cell vortex flow occurs. The dependence on a cylindrical channel with aspect ratio η = ri/ro = 0.9. the aspect ratio η is shown in Fig. 5. As can be seen, As one can see, the relative error decreases rapidly when ec2 decreases monotonically with the aspect ratio. For the grid resolution is increased. This shows that, within veryD narrow channels with aspect ratio η > 0.95, the an error of 1%, our numerical results correspond to the second bifurcation occurs already at relatively low Dean physical values and are not affected by finite resolution numbers between 45 and 50. effects. 7

FIG. 6. (Color online) Relative error of the critical Dean number as function of the number of grid points for a cylindrical channel with aspect ratio η = ri/ro = 0.9.

FIG. 7. (Color online) Geometry of the double-curved chan- IV. FLOW THROUGH DOUBLE-CURVED nel. The colored spots on the channel cross-section at the CHANNEL inlet illustrate the strength of the ﬂow vorticity. The dashed lines indicate the periodicity of the channel in φ-direction.

We now consider a more complex geometry, namely a double-curved channel (see Fig. 7). This geometry fol- with non-vanishing Christoffel symbols lows from the cylindrical configuration by introducing a Γr = r cos2 φ, second curvature along the z-axis, which leads to a ge- θθ − Γr = r, ometry that can best be described in terms of ellipsoidal φφ − coordinates (x1, x2, x3) = (r, θ, φ), defined by θ θ φ φ Γrθ =Γθr =Γrφ =Γφr =1/r, Γθ =Γθ = tan φ, θφ φθ − x = ra cos θ cos φ, φ Γθθ = sin φ cos φ. y = rb sin θ cos φ, In our simulations, we use a D3Q41 lattice of size L z = rc sin φ, r × L L = 64 64 128. The lattice indices (ˆr, θ,ˆ φˆ) θ × φ × × ∈ [1,Lr] [1,Lθ] [1,Lφ] are related to the physical units by × × where a,b,c are the lengths of the three semi-principle axes of the ellipsoid. All vector fields and tensors are ˆ Lθ ˆ Lφ expressed in terms of the basis (e , e , e ) = ( ∂ , ∂ , ∂ ). r = ri +ˆr ∆r, θ = θ ∆θ, φ = φ ∆φ, r θ φ ∂r ∂θ ∂φ · − 2 − 2 Since finite-size effects in channels of finite height are and, as in the cylindrical case, we choose ∆r = ∆θ = known to have a strong influence on the Dean instabil- ∆φ = ∆t = (1 η)/(ηLr), where η = ri/ro denotes the ity (as studied in Ref. [6] for the case of two concentric aspect ratio of− the channel. cylinders), we impose periodicity along the φ-direction by The inner and outer walls of the channel at r = r choosing a = c = 1. This corresponds to the periodicity i and r = ro are implemented in the same way as for the in z-direction for the cylindrical case and avoids finite- cylindrical case by using Dirichlet boundary conditions height complications. The corresponding metric compo- for the fluid velocity, u(ro) = u(ri) = 0. We further use nents are given by open boundary conditions in θ-direction as well as periodic boundaries in φ. Since by construction the channel

2 2 2 2 2 is periodic in φ, we can restrict the simulation domain to grr = sin φ + cos φ cos θ + b sin θ , a circular sector in φ (as indicated by the dotted lines in 2 2 2 2 2 gθθ = r cos φ sin θ + b cos θ , Fig. 7). Note that by restricting to a finite sector and considering periodic boundary conditions, a symmetry g = r2 cos2 φ+ sin2 φ cos2 θ + b2 sin2 θ , φφ condition is imposed on the solution, which might ex- 2 2 grθ = gθr = r b 1 cos φ cos θ sin θ, clude some phases in the bifurcation diagram. However, − 2 2 since this is also the case for the cylindrical channel to grφ = gφr = r 1 b sin θ cos φ sin φ, − which we want to compare, we do not take those phases g = g = r2 1 b2 cos φ sin φ cos θ sin θ, θφ φθ − into account. In particular, the double-curved channel is 8 constructed in such a way that it approaches the geome- for b = 0.9. As can be observed, the vorticity begins to try of the cylindrical channel (Fig. 1) for increasing inner increase at Dean number ec 30, which indicates an Lθ D ≈ radii ri. The open channel boundaries at θ = 2 ∆θ de- imperfect bifurcation from Poiseuille flow to 2-cell vortex fine an inlet and an outlet for the flow, as can± be seen in flow. Compared to the cylindrical channel, this transition Fig. 7. In order to drive the fluid through the channel, is however rather smooth. The colored pictures in Fig. 8 an external force in θ-direction can be applied, which for depict the vorticity ωθ on a cross-section. incompressible flow is physically equivalent to a pressure We have measured the critical Dean number at the bi- gradient between the inlet and the outlet. furcation point for different streamwise curvatures θ = 2 R We study the bifurcation from curved channel rib . The results are depicted in Fig. 9, where errorbars Poiseuille flow to Dean vortex flow by varying the Dean represent the uncertainty in reading off the critical Dean number, number from the vorticity curve. Two bifurcation points have been observed: a bifurcation from Poiseuille flow to vθ d d 2-cell vortex flow for > 0.7 and a bifurcation from e = h i , (9) Rθ D ν · r θ Poiseuille flow to 4-cell vortex flow for θ < 0.7. Fig. 9 R shows that the critical Dean number forR the bifurcation where d = r r and ν = (τ 1/2) c2∆t. denotes o − i − s Rθ to 2-cell vortex flow possesses a minimum at a stream- the streamwise curvature radius of the inner wall, given wise curvature θ = 1.00 0.05, which corresponds to by = r b2 at θ = 0. The mean azimuthal velocity v R ± Rθ i h θi the spherical geometry, where all semi-principal axes of is calculated as follows, the ellipsoid are equal. For θ = 1.00 0.05, the instability occurs already at a relativelyR low± Dean number of 1 θ 2 vθ = u r b cos φ drdφ, ec 22.5. h i S Z D ≈ S For θ > 1, the geometry of the channel is equivalent toR a streamwise stretched ellipsoid, such that the where S = r drdφ is the cross-sectional area of the streamwise curvature is expected to have only a minor channel and uθ is the azimuthal component of the veloc- R influence on the instability relative to the spherical case ity field u = ure + uθe + uφe . Like in the cylindrical r θ φ = 1.00 0.05. Still, the critical Dean number in- case, we measure the average vorticity in the streamwise θ Rcreases almost± linearly with for > 1, which sug- direction at θ = 0, given by θ θ gests that the second curvatureR dominatesR the insta- Rφ 1 ∂ur ∂uφ bility in this regime. ωθ = rb cos φ drdφ. For < 1, on the other hand, the geometry of the h i S Z ∂φ − ∂r θ S channelR resembles an ellipsoid compressed in streamwise

We study two different cases: In the first case, we vary direction, and the instability is dominated by the increasing streamwise curvature. For < 1, we have observed the streamwise curvature radius of the channel θ = Rθ 2 R the following behavior of the flow next to the bifurcation rib by changing the length of the semi-principal axes point: Right before the vortices begin to develop, there b while the inner radius ri is kept fixed. In the second case, we study the effect of the cross-sectional curvature are four regions of slightly increased vorticity: Two next to the center of the outer wall at r and two next to the radius φ = ri along the φ direction by varying the inner o R corners of the inner wall at r . This can be seen in the radius ri while keeping the streamwise curvature radius i lower colored picture in Fig. 8, next to the bifurcation θ fixed. R point. For 0.7 < θ < 1, the two inner regions domi- nate and finally formR two major vortices (see the upper A. Variation of streamwise curvature colored picture in Fig. 8). For θ < 0.7, on the other hand, the two vorticity regions inR the corner also grow in First, we vary the curvature radius along the flow direc- strength and, together with the two inner regions, they tion, = r b2, by varying b at fixed cross-sectional cur- finally form four major vortices (see the left colored pic- Rθ i ture in Fig. 9). This behavior is different from the case vature radius φ = ri = 1. For the channel, we choose R θ > 1, where we observe only two regions of increased an aspect ratio of η = ri/ro =0.9. For all the simulations R of the double-curved channel, we set the relaxation time vorticity in the beginning, which finally form two major τ to 1 in order to work in the desired parameter range of vortices. the Reynolds number (and Dean number), keeping at the same time a good computational performance. The fluid is initialized with a uniform mass distribution by setting B. Variation of cross-sectional curvature ρ = ρ′√g = 1 at t = 0. The Dean number is varied by changing the strength of the driving force. We also have studied the effect of the cross-sectional Again, we plot the average vorticity ωθ as function curvature radius = r on the Dean instability by vary- h i Rφ i of the Dean number e in order to determine the crit- ing the inner radius ri of the channel. The streamwise ical Dean number, atD which the vorticity suddenly in- curvature radius = r b2 is set to 1 and is kept fixed Rθ i creases. Exemplarily, Fig. 8 shows the vorticity curve in all simulations by setting b =1/√ri. The aspect ratio 9

the cross-sectional curvature 1/ φ for φ < 1. This suggests that for < 1 the instabilityR R is dominated Rφ by the cross-sectional curvature 1/ φ. For φ > 1 on the other hand, the critical Dean numberR growsR almost linearly from 22.5 at = 1 to a value of about 35 at Rφ φ =1.2, while a second bifurcation from 2-cell to 4-cell vortexR flow begins to appear at higher Dean numbers. In this range, the instability is dominated by the perpendicular streamwise curvature 1/ θ. For φ > 1.2, the critical Dean number of the firstR bifurcationR stays more or less constant, whereas the threshold for the second bi- FIG. 8. (Color online) Average vorticity ωθ as function of furcation decreases further and further until it reaches a the Dean number e for a double-curvedh channeli with inner minimum at φ =1.55 0.05. At this point, the third bi- D R ± radius ri = 1, aspect ratio η = ri/ro = 0.9 and b = 0.9. furcation from 4-cell to 6-cell vortex flow appears, while The bifurcation from Poiseuille flow to vortex flow occurs at the critical Dean number at the first bifurcation point e 30.5. The colored pictures depict the vorticity on a approaches the corresponding value for the cylindrical D ≈ channel cross-section, where the blue (right) and red (left) channel in the limit φ . spots correspond to clockwise and counterclockwise rotating From the qualitativeR point→ ∞ of view, the three curves in vortices, respectively. Fig. 10 show a similar behaviour: Starting from different values of φ, within error bars all curves decrease to a minimumR value right before the next bifurcation emerges. When the next bifurcation appears, the critical Dean number of the lower bifurcation increases to a rather constant value, which grows only slightly towards φ . R Physically,→ ∞ the appearance of 4-cells or 6-cells can be explained by the splitting mechanism described in Ref. [18]: By increasing the Dean number, the Dean vortices can split up and form new vortex pairs. Although, one might wonder why there is no direct bifurcation from 4 cells to 8 cells, since in theory, the 4 vortices should be FIG. 9. (Color online) Critical Dean number at the bifurca- 2 completely indistinguishable. In practice, however, the tion points versus the streamwise curvature radius θ = rib R symmetry between the cells is broken by small pertur- for a double-curved channel with aspect ratio η = ri/ro = 0.9. bations, which leads to the splitting of only one vortex The colored pictures depict the streamwise vorticity of the pair, resulting in the 6-cell solution. From the numeri- flow on a channel cross-section. cal point of view, we have checked the physicality of the 6-cell solution by doubling the grid resolution as well as of the channel is set to η = 0.9, and the lattice spacing by changing the triggering mechanism, finding the same result in both cases. Exemplarily, Fig. 11 shows the in radial direction is given by ∆r = (1 η)/(ηLr). In order to keep the physical dimensions of− the channel fixed cross-sectional velocity profile of a 6-cell vortex flow for φ =1.8 and e = 44. when ri varies, the lattice spacings in θ- and φ-direction R D are rescaled accordingly: ∆θ = ∆r/(b ri), ∆φ = ∆r/ri. Like in previous studies, we have analyzed the vorticity curve as function of the Dean number. Depending on V. CONCLUSIONS the cross-sectional curvature radius φ, we have found three different bifurcations: a bifurcationR from Poiseuille Summarizing, we have studied the Dean instability in flow to 2-cell vortex flow, from 2 cells to 4 cells as well as a double-curved channel, using our previously developed from 4 cells to 6 cells. Fig. 10 depicts the dependence of LB algorithm in general coordinates. The double-curved the critical Dean number at the transition points on the channel is characterized by a streamwise curvature as well curvature radius φ. The first bifurcation is the bifurca- as by a perpendicular cross-sectional curvature. In anal- tion from curvedR channel Poiseuille flow to 2-cell vortex ogy with cylindrical channels, which have been widely flow. As can be observed, the critical Dean number for studied in the past, we have observed a bifurcation from this transition possesses a minimum at φ =1.00 0.05, primary curved channel Poiseuille flow to secondary Dean corresponding to a spherical channel geometry,R and± the vortex flow, which is characterized by counter-rotating minimum value of the critical Dean number, ec 22.5 vortices. In particular, we have measured the critical at = 1.00 0.05 coincides with the correspondingD ≈ Dean number at the bifurcation points as function of the Rφ ± value in Fig. 9 for the spherical case ( θ = 1). As can be geometrical properties of the channel. seen in Fig. 10, the critical Dean numberR increases with At first, we have varied the streamwise curvature ra- 10

stronger streamwise curvature, the critical Dean number increases almost linearly. For strongly-curved channels, we even have observed bifurcations from Poiseuille flow to 4-cell vortex flow. Secondly, we also have varied the cross-sectional curvature radius φ while keeping the streamwise curvature fixed. Again,R we have found that the lowest Dean number at which the Dean instability occurs corresponds to the spherically symmetric configuration, in which both curvature radii are approximately equal, θ φ, as before. When the cross-sectional curvatureR decreases≈ R towards the cylindrical channel limit, φ , higher order bifurcations from 2-cell flow to 4-cellR → flow ∞ and even FIG. 10. (Color online) Critical Dean number at the bifurca- from 4-cell flow to 6-cell vortex flow come into play, while tion points as function of the cross-sectional curvature radius the critical Dean number of the first bifurcation from φ = ri. The colored pictures depict the streamwise vorticity ofR the flow on a channel cross-section. Poiseuille flow to 2-cell flow approaches the corresponding value for the cylindrical channel.

radial velocity ur For the simulations, we have improved our previously 0.020 developed LB algorithm in general coordinates to sim- 0.000 ulate flow in curved channels with complex geometries. The improved LB method has been validated for the case −0.020 of flow through a cylindrical channel, for which we have measured the critical Dean number at the transition from streamwise velocity uθ laminar flow to Dean vortex flow for different aspect ra- 0.450 tios of the channel. The results agree very well with 0.400 numerically obtained results by Finlay et al. [6]. In ad- dition to the linear stability analysis by Finlay et al., we 0.350 have observed a second bifurcation from 2-cell to 4-cell vortex flow, supporting the existence of a second critical azimuthal velocity u 0.004 φ Dean number at the second bifurcation point. The double-curved channel is implemented using an 0.000 ellipsoidal coordinate system, which enters in our algorithm simply through the metric tensor. By using con- −0.004 travariant coordinates, the LB equation automatically adapts to the new geometry, which demonstrates the 0 0.2 0.4 0.6 0.8 1 great advantage of our method when dealing with com- φ/Lφ plex three-dimensional geometries. As we use generalized coordinates which are perfectly adapted to the channel FIG. 11. (Color online) Velocity profile of 6-cell vortex flow in geometry, the channel boundaries can be implemented a double-curved channel with aspect ratio η = 0.9, streamwise 2 accurately without using the staircase approximation. curvature θ = rib = 1 and cross-sectional curvature φ = R R ri = 1.8. The corresponding Dean number is e = 44. D dius while keeping the cross-sectional curvature ra- ACKNOWLEDGMENTS Rθ dius φ fixed. We have found that the critical Dean number atR the bifurcation from Poiseuille flow to 2-cell vor- We acknowledge financial support from the Euro- tex flow is minimal for θ φ, where the channel pos- pean Research Council (ERC) Advanced Grant 319968- sesses a spherical symmetry.R ≈ For R channels with weaker or FlowCCS.

[1] W. H. Finlay, J. B. Keller, and J. H. Ferziger, Journal ysis 1, 212 (1957). of Fluid Mechanics 194, 417 (1988). [4] D. B. Brewster, P. Grosberg, and A. H. Nissan, Pro- [2] W. Dean, Proceedings of the Royal Society of London. ceedings of the Royal Society of London. Series A. Math- Series A 121, 402 (1928). ematical and Physical Sciences 251, 76 (1959). [3] G. H¨ammerlin, Archive for rational mechanics and anal- [5] W. H. Finlay, Physics of Fluids A: Fluid Dynamics (1989- 11

1993) 1, 854 (1989). reports 3, 3106 (2013). [6] W. Finlay and K. Nandakumar, Physics of Fluids A: [13] M. Mendoza, J.-D. Debus, S. Succi, and H. Herrmann, Fluid Dynamics (1989-1993) 2, 1163 (1990). International Journal of Modern Physics C (2014). [7] S. Chen and G. D. Doolen, Annual review of ﬂuid me- [14] J. G. Zhou, Phys. Rev. E 78, 036701 (2008). chanics 30, 329 (1998). [15] P. L. Bhatnagar, E. P. Gross, and M. Krook, Phys. Rev. [8] M. Mendoza and J. D. Munoz, Physical Review E 82, 94, 511 (1954). 056708 (2010). [16] S. Chapman and T. Cowling, Cambridge, UK (1970). [9] M. Mendoza and J. D. Munoz, Physical Review E 77, [17] S. S. Chikatamarla and I. V. Karlin, Physical Review E 026713 (2008). 79, 046701 (2009). [10] M. Mendoza, B. M. Boghosian, H. J. Herrmann, and [18] Y. Guo and W. Finlay, Journal of Fluid Mechanics 228, S. Succi, Physical review letters 105, 014502 (2010). 661 (1991). [11] F. Mohseni, M. Mendoza, S. Succi, and H. J. Herrmann, [19] P. E. Haines, J. P. Denier, and A. P. Bassom, Journal Phys. Rev. D 87, 083003 (2013). of Fluid Mechanics 716, R8 (2013). [12] M. Mendoza, S. Succi, and H. J. Herrmann, Scientiﬁc