Local and Nonlocal Strain Rate Fields and Vorticity Alignment in Turbulent Flows
aa r X i v : . [ phy s i c s . f l u - dyn ] J a n Local and Nonlocal Strain Rate Fields and Vorticity Alignment in Turbulent Flows
Peter E. Hamlington ∗ , J¨org Schumacher † , and Werner J. A. Dahm ‡ Laboratory for Turbulence & Combustion (LTC), Department of Aerospace Engineering,The University of Michigan, Ann Arbor, MI 48109-2140, USA Department of Mechanical Engineering, Technische Universit¨at Ilmenau, D-98684 Ilmenau, Germany (Dated: October 25, 2018)Local and nonlocal contributions to the total strain rate tensor S ij at any point x in a flow areformulated from an expansion of the vorticity field in a local spherical neighborhood of radius R centered on x . The resulting exact expression allows the nonlocal (background) strain rate tensor S Bij ( x ) to be obtained from S ij ( x ). In turbulent flows, where the vorticity naturally concentrates intorelatively compact structures, this allows the local alignment of vorticity with the most extensionalprincipal axis of the background strain rate tensor to be evaluated. In the vicinity of any vorticalstructure, the required radius R and corresponding order n to which the expansion must be carriedare determined by the viscous lengthscale λ ν . We demonstrate the convergence to the backgroundstrain rate field with increasing R and n for an equilibrium Burgers vortex, and show that thisresolves the anomalous alignment of vorticity with the intermediate eigenvector of the total strainrate tensor. We then evaluate the background strain field S Bij ( x ) in DNS of homogeneous isotropicturbulence where, even for the limited R and n corresponding to the truncated series expansion, theresults show an increase in the expected equilibrium alignment of vorticity with the most extensionalprincipal axis of the background strain rate tensor. PACS numbers: 47.27.-i,47.32.C-,47.27.De
I. INTRODUCTION
Vortex stretching is the basic mechanism by which ki-netic energy is transfered from larger to smaller scales inthree-dimensional turbulent flows [1, 2, 3, 4]. An under-standing of how vortical structures are stretched by thestrain rate field S ij ( x ) is thus essential to any descriptionof the energetics of such flows. Over the last two decades,direct numerical simulations (DNS) [5, 6, 7] and experi-mental studies [8, 9, 10, 11, 12] of the fine-scale structureof turbulence have revealed a preferred alignment of thevorticity with the intermediate eigenvector of the strainrate tensor. This result has been widely regarded as sur-prising. Indeed the individual components of the inviscidvorticity transport equation, in a Lagrangian frame thatremains aligned with the eigenvectors of the strain ratetensor, are simplyD ω D t = s ω , D ω D t = s ω , D ω D t = s ω , (1)where s , s and s are the eigenvalues of S ij . For in-compressible flow, s + s + s ≡
0, and then denoting s ≥ s ≥ s requires s ≥ s ≤
0. As a con-sequence, (1) would predict alignment of the vorticitywith the eigenvector corresponding to the most exten-sional principal strain rate s . Yet DNS and experimen-tal studies have clearly shown that the vorticity insteadis aligned with the eigenvector corresponding to the in-termediate principal strain rate s . ∗ email: [email protected] (corresponding author) † email: [email protected] ‡ email: [email protected] A key to understanding this result is that, owing tothe competition between strain and diffusion, the vortic-ity in turbulent flows naturally forms into concentratedvortical structures. It has been noted, for example inRefs. [7, 13, 14], that the anomalous alignment of thevorticity with the strain rate tensor S ij ( x ) might be ex-plained by separating the local self-induced strain ratefield created by the vortical structures themselves fromthe background strain field in which these structures re-side. The total strain rate tensor is thus split into S ij ( x ) = S Rij ( x ) + S Bij ( x ) , (2)where S Rij is the local strain rate induced by a vorticalstructure in its neighboring vicinity, and S Bij is the non-local background strain rate induced in the vicinity ofthe structure by all the remaining vorticity. The vorticalstructure would then be expected to align with the princi-pal axis corresponding to the most extensional eigenvalueof the background strain rate tensor S Bij ( x ).In the following, we extend this idea and suggest asystematic expansion of the total strain rate field S ij ( x )that allows the background strain rate field S Bij ( x ) tobe extracted. Our approach is based on an expansionof the vorticity over a local spherical region of radius R centered at any point x . This leads to an exact opera-tor that provides direct access to the background strainrate field. The operator is tested for the case of a Burg-ers vortex, where it is shown that the local self-inducedstrain field produced by the vortex can be successfully re-moved, and the underlying background strain field can beincreasingly recovered as higher order terms are retainedin the expansion. The anomalous alignment of the vor-ticity with respect to the eigenvectors of the total strainfield is shown in that case to follow from a local switch- FIG. 1: Decomposition of the vorticity field in the vicinityof any point x into local and nonlocal parts; the Biot-Savartintegral in (8) over each part gives the local and nonlocal(background) contributions to the total strain rate tensor S ij at x . ing of the principal strain axes when the vortex becomessufficiently strong relative to the background strain. Fi-nally, the operator is applied to obtain initial insightsinto the background strain S Bij ( x ) in DNS of homoge-neous isotropic turbulence, and used to compare the vor-ticity alignment with the eigenvectors of the total strainfield and of this background strain field. II. THE BACKGROUND STRAIN FIELD
The velocity u at any point x induced by the vorticityfield ω ( x ) is given by the Biot-Savart integral u ( x ) = 14 π Z Λ ω ( x ′ ) × x − x ′ | x − x ′ | d x ′ , (3)where the integration domain Λ is taken to be infinite orperiodic. In index notation (3) becomes u i ( x ) = 14 π Z Λ ǫ ilk ω l ( x ′ ) ( x k − x ′ k ) | x − x ′ | d x ′ , (4)where ǫ ilk is the cyclic permutation tensor. The deriva-tive with respect to x j gives the velocity gradient tensor ∂∂x j u i ( x ) = 14 π Z Λ ǫ ilk ω l ( x ′ ) (cid:20) δ kj r − r k r j r (cid:21) d x ′ , (5)where r ≡ | x − x ′ | and r m ≡ x m − x ′ m . The strainrate tensor S ij at x is the symmetric part of the velocitygradient, namely S ij ( x ) ≡ (cid:18) ∂u i ∂x j + ∂u j ∂x i (cid:19) . (6) From (5) and (6), S ij ( x ) can be expressed [15] as anintegral over the vorticity field as S ij ( x ) = 38 π Z Λ ( ǫ ikl r j + r i ǫ jkl ) r k r ω l ( x ′ ) d x ′ . (7)As shown in Fig. 1, the total strain rate in (7) is separatedinto the local contribution induced by the vorticity withina spherical region of radius R centered on the point x and the remaining nonlocal (background) contributioninduced by all the vorticity outside this spherical region.The strain rate tensor in (7) thus becomes S ij ( x ) = 38 π Z r ≤ R [ · · · ] d x ′ | {z } + 38 π Z r>R [ · · · ] d x ′ | {z } , (8) ≡ S Rij ( x ) ≡ S Bij ( x )where [ · · · ] denotes the integrand in (7). The nonlocalbackground strain tensor at x is then S Bij ( x ) = S ij ( x ) − S Rij ( x ) . (9)The total strain tensor S ij ( x ) in (9) is readily evaluatedvia (6) from derivatives of the velocity field at point x .Thus all that is required to obtain the background (non-local) strain rate tensor S Bij ( x ) via (9) is an evaluationof the local strain integral S Rij ( x ) in (8) produced by thevorticity field ω l ( x ′ ) within r ≤ R in Fig. 1. A. Evaluating the Background Strain Rate Tensor
The vorticity field within the sphere of radius R canbe represented by its Taylor expansion about the centerpoint x as ω l ( x ′ ) | r ≤ R = ω l ( x ) + ( x ′ m − x m ) ∂ω l ∂x m (cid:12)(cid:12)(cid:12)(cid:12) x (10)+ 12 ( x ′ m − x m ) ( x ′ n − x n ) ∂ ω l ∂x m ∂x n (cid:12)(cid:12)(cid:12)(cid:12) x + · · · . Recalling that x m − x ′ m ≡ r m and using a l , b lm , c lmn , . . . to abbreviate the vorticity and its derivatives at x , wecan write (10) as ω l ( x ′ ) | r ≤ R ≡ a l − r m b lm + 12 r m r n c lmn − · · · . (11)Substituting (11) in the S Rij integral in (8) and changingthe integration variable to r = x − x ′ , the strain tensorat x produced by the vorticity in R is S Rij ( x ) = 38 π Z r ≤ R ( ǫ ikl r j + r i ǫ jkl ) r k r (12) × h a l − r m b lm + r m r n c lmn − · · · i d r . This integral can be solved in spherical coordinates cen-tered on x , with r = r sin θ cos φ , r = r sin θ sin φ , and r = r cos θ for r ∈ [0 , R ], θ ∈ [0 , π ], and φ ∈ [0 , π ). Tointegrate (12) note that Z r ≤ R r k r j r d r = 4 π δ jk Z R r dr , (13a) Z r ≤ R r k r j r m r d r = 0 , (13b) Z r ≤ R r k r j r m r n r d r = 2 π R ( δ mn δ jk + (13c) δ mj δ kn + δ mk δ jn ) . The resulting local strain rate tensor at x is then S Rij ( x ) = R c lmn ( ǫ ijl δ mn + ǫ jil δ mn + ǫ inl δ mj (14)+ ǫ jnl δ mi + ǫ iml δ nj + ǫ jml δ ni ) + O ( R ) , where the contribution from the a l term in (12) is zerosince ǫ ijl = − ǫ jil . For the same reason the first two termsin (14) also cancel, giving S Rij ( x ) = R c lmn ( ǫ inl δ mj + ǫ jnl δ mi (15)+ ǫ iml δ nj + ǫ jml δ ni ) + O ( R ) . Recalling that c lmn = c lnm ≡ ∂ ω l /∂x m ∂x n , and con-tracting with the δ and ǫ in (15), gives S Rij ( x ) = R (cid:20) ∂∂x j (cid:18) ǫ iml ∂ω l ∂x m (cid:19) (16)+ ∂∂x i (cid:18) ǫ jml ∂ω l ∂x m (cid:19)(cid:21) + O ( R ) . Note that ǫ iml ∂ω l /∂x m ≡ ( ∇ × ω ) i and ∇ × ω = ∇ × ( ∇ × u ) = ∇ ( ∇ · u ) − ∇ u , (17)so for an incompressible flow ( ∇ · u ≡
0) the local strainrate tensor at x becomes S Rij ( x ) = − R ∇ (cid:18) ∂u i ∂x j + ∂u j ∂x i (cid:19) + O ( R ) . (18)From (9), with S Rij from (18) we obtain the backgroundstrain tensor as S Bij ( x ) = S ij ( x ) + R ∇ S ij ( x ) + O ( R ) . (19)The remaining terms in (19) result from the higher-orderterms in (11). The contributions from each of these canbe evaluated in an analogous manner, giving S Bij ( x ) = (cid:20) R ∇ + R ∇ ∇ + · · · (20)+ 3 R n − (2 n − n − (cid:0) ∇ (cid:1) n − + · · · (cid:21) S ij ( x ) , where the terms shown in (20) correspond to n = 1 , , . . . .The final result in (20) is an operator that extracts thenonlocal background strain rate tensor S Bij at any point x from the total strain rate tensor S ij . For the Taylorexpansion in (10), this operator involves Laplacians ofthe total strain rate field S ij ( x ). B. Practical Implementation
When using (20) to examine the local alignment of anyconcentrated vortical structure with the principal axes ofthe background strain rate field S Bij ( x ) in which it resides,the radius R must be taken sufficiently large that thespherical region | x ′ − x | ≤ R encloses essentially all thevorticity associated with the structure, so that its localinduced strain rate field is fully accounted for. Gener-ally, as R increases it is necessary in (20) to retain termsof increasingly higher order n to maintain a sufficientrepresentation of ω ( x ′ ) over the spherical region. Thusfor any vortical structure having a characteristic gradi-ent lengthscale λ ν , it can be expected that R must be ofthe order of λ ν , and n will then need to be sufficientlylarge to adequately represent the vorticity field withinthis sphere. However, since the local gradient lengthscalein the vorticity field in a turbulent flow is determined byan equilibrium between strain and diffusion, the vortic-ity field over the lengthscale λ ν will be relatively smooth,and thus relatively low values of n may suffice to give ausable representation of ω ( x ′ ). This is examined in thefollowing Section. III. TEST CASE: BURGERS VORTEX
The equilibrium Burgers vortex [1, 3, 9, 16] is formedfrom vorticity in a fluid with viscosity ν by a spatiallyuniform, irrotational, axisymmetric background strainrate field S Bij that has a single extensional principal strainrate S zz directed along the z axis, as shown in Fig. 2.This simple flow, often regarded as an idealized modelof the most concentrated vortical structures in turbulentflows, provides a test case for the result in (20). Thecombined strain rate field S ij ( x ) produced by the vor-tex and the background strain flow should, when applied FIG. 2: Equilibrium Burgers vortex with circulation Γ andstrain-limited viscous diffusion lengthscale λ ν in a uniform,irrotational, axisymmetric background strain rate field S Bij ( x ). in (20), produce the underlying background strain field( S Brr , S
Bθθ , S
Bzz ) = ( − , − , S zz at all x when R → ∞ and all orders n are retained. For finite R and n , theresulting S Bij ( x ) will reflect the convergence properties of(20). A. Strain Rate Tensor
The equilibrium Burgers vortex aligned with the exten-sional principal axis of the background strain rate fieldhas a vorticity field ω ( x ) = ω z ( r )ˆ z = απ Γ λ ν exp (cid:0) − αη (cid:1) ˆ z , (21)where Γ is the circulation, λ ν is the viscous lengthscalethat characterizes the diameter of the vortex, η ≡ r/λ ν is the radial similarity coordinate, and the constant α reflects the chosen definition of λ ν . Following [9], λ ν istaken as the full width of the vortical structure at which ω z has decreased to one-fifth of its peak value, for which α = 4 ln 5. When diffusion of the vorticity is in equilib-rium [9] with the background strain, then λ ν = √ α (cid:18) νS zz (cid:19) / . (22)The combined velocity field u ( x ) produced by the vortexand the irrotational background strain is given by thecylindrical components u r ( r, θ, z ) = − S zz r , (23a) u θ ( r, θ, z ) = Γ2 πλ ν η (cid:2) − exp (cid:0) − αη (cid:1)(cid:3) , (23b) u z ( r, θ, z ) = S zz z . (23c)The combined strain rate tensor for such a Burgersvortex is thus S ij ( x ) = − S zz / S vrθ S vrθ − S zz / S zz , (24)where S vrθ is the shear strain rate induced by the vortex,given by S vrθ ( x ) = Γ πλ ν (cid:20)(cid:18) α + 1 η (cid:19) exp (cid:0) − αη (cid:1) − η (cid:21) . (25)From (24), S ij ( x ) has one extensional principal strainrate equal to S zz along the ˆ z axis, with the remainingtwo principal strain axes lying in the r - θ plane and cor-responding to the principal strain rates s = − S zz ± | S vrθ | . (26)As long as the largest s in (26) is smaller than S zz , themost extensional principal strain rate s of S ij will be S zz , and the corresponding principal strain axis will pointin the ˆ z direction. The vorticity is then aligned with themost extensional eigenvector of S ij . This remains thecase until the vortex becomes sufficiently strong relativeto the background strain rate that s > S zz , namely | S vrθ | ≥ S zz , (27)which from (25) occurs wherever (cid:18) α + 1 η (cid:19) exp( − αη ) − η ≥ π (cid:18) Γ /λ ν S zz (cid:19) − . (28)At any η for which (28) is satisfied, the most extensionalprincipal axis of the combined strain rate tensor S ij ( x )will switch from the ˆ z direction to instead lie in the r - θ plane. Since the vorticity vector everywhere points inthe ˆ z direction, wherever (28) is satisfied the principalaxis of S ij that is aligned with the vorticity will switchfrom the most extensional eigenvector to the intermediateeigenvector. This alignment switching is purely a resultof the induced strain field S vij ( x ) locally dominating thebackground strain field S Bij ( x ).The dimensionless vortex strength parameterΩ ≡ (cid:20) Γ /λ ν S zz (cid:21) = πα ω max S zz (29)on the right-hand side of (28) characterizes the relativestrength of the background strain and the induced strainfrom the vortical structure, where ω max is obtained from(21) at η = 0. For Ω < Ω ∗ ≈ . , (30)the background strain rate S zz is everywhere larger thanthe largest s in (26), and thus no alignment switchingoccurs at any η . For Ω > Ω ∗ , alignment switching willoccur over the limited range of η values that satisfy (28).With increasing values of Ω, more of the vorticity fieldwill be aligned with the intermediate principal axis ofthe combined strain rate tensor, even though all of thevorticity field remains aligned with the most extensionalprincipal axis of the background strain rate tensor.Figure 3 shows the vorticity ω z and the induced shearstrain component − S vrθ as a function of η . The horizontaldashed lines correspond to three different values of Ω,and indicate the range of η values where the alignmentswitching in (28) occurs for each Ω. Wherever − S vrθ isabove the dashed line for a given Ω, the vorticity will bealigned with the local intermediate principal axis of thecombined strain rate field.In principle, regardless of the vortex strength parame-ter Ω, at any η the result in (20) can reveal the alignmentof the vorticity with the most extensional principal axisof the background strain field S Bij . However, this requires R to be sufficiently large that a sphere with diameter 2 R ,centered at the largest η for which − S vrθ in Fig. 3 is still (cid:358) (cid:358) (cid:358) ω z ω max η (cid:58) = (cid:58) *=2.452 (cid:58) *3 (cid:58) * − S rθ ω max (cid:358) S r (cid:84) (cid:90) z FIG. 3: Similarity profiles of ω z ( η ) and S rθ ( η ) for any equi-librium Burgers vortex; wherever − S rθ exceeds the horizontalline determined by the relative vortex strength parameter Ωin (29) the most extensional principal axis of the total strainrate S ij ( x ) switches from the ˆ z -axis to lie in the r - θ plane. above the horizontal dashed line, will enclose essentiallyall of the vorticity associated with the vortical structure.As Ω increases, the required R will increase accordinglyas dictated by (28), and as R is increased the required n in (20) also increases.Irrespective of the value of Ω, when (20) is applied tothe combined strain rate field S ij ( x ) in (24) and (25),if ˜ R ≡ ( R/λ ν ) → ∞ and all orders n are retained thenthe resulting S Bij ( x ) should recover the background strainfield, namely S Brθ → x , and the vorticity should show alignment withthe most extensional principal axis of S Bij . For finite
R/λ ν and various orders n , the convergence of S Bij from (20) tothis background strain field is examined below.
B. Convergence of the Background Strain
The accuracy with which (20) can recover the back-ground strain field S Bij ( x ) that acts on a concentratedvortical structure depends on how well the expansion in(10) represents the vorticity field within the local spheri-cal neighborhood R . Figure 4 shows the results of a localsixth-order Taylor series approximation for the vorticityin (21) at various radial locations across the Burgers vor-tex. In each panel, the blue square marks the location x at which the sphere is centered, and the red dashed curveshows the resulting Taylor series approximation for thevorticity. On the axis of the vortex, the approximatedvorticity field correctly accounts for most of the circula-tion in the vortex, and thus the induced strain field fromthe vortex will be reasonably approximated. Off the axis, the approximation becomes increasingly poorer, but the1 /r decrease in the Biot-Savart kernel in (3) neverthe-less renders it adequate to account for most of the vortex-induced strain rate field. At the largest radial location,corresponding to the bottom right panel of Fig. 4, the ap-proximation becomes relatively poor, however at large η values the vortex-induced strain is sufficiently small thatit is unlikely to lead to alignment switching for typical Ωvalues.Figure 5 shows the shear component S Brθ ( η ) of thebackground strain rate tensor obtained via (20) for var-ious n and ˜ R as a function of η . In each panel, theblack curve shows the total strain rate S ij ( η ) and the col-ored curves show the background strain rate S Bij ( η ) from(20) for the ( n, ˜ R ) combinations listed. The horizontaldashed line corresponding to Ω = (3 / ∗ reflects the rel-ative vortex strength, and shows the range of η where theanomalous alignment switching occurs due to the vortex-induced strain field. Wherever the − S Brθ curves are abovethis line, the vorticity there will be aligned with the inter-mediate principal axis of the combined strain rate tensor S ij . Figure 5( a ) examines the effect of increasing theradius ˜ R of the spherical region for fixed order n = 6.It is apparent that with increasing ˜ R the resulting − S Brθ converges toward the correct background strain field in(31). For the value of Ω shown, it can be seen that for R & . λ ν the resulting S Brθ is everywhere below the hor-izontal dashed line, indicating that the vorticity every-where is aligned with the most extensional principal axisof the resulting background strain rate tensor S Bij ( x ) from(20). (cid:358) (cid:358) (cid:358) (cid:358) ω z ω max (cid:358) (cid:358) ω z ω max η (cid:358) (cid:358) η FIG. 4: Accuracy of the Taylor expansion for the local vortic-ity in (10) for a Burgers vortex, showing results for 6th orderapproximation. In each panel, solid black curve shows actualvorticity profile, and red dashed curve gives approximatedvorticity from derivatives at location marked by square. (cid:358) (cid:358) (a) η − S Brθ ω max π α Ω (cid:358) S r (cid:84) (6, 0.20)(6, 0.35)(6, 0.50)(6, 0.65) (cid:358) (cid:358) (cid:358) (b) η − S Brθ ω max π α Ω (cid:358) S r (cid:84) (3, 0.65)(4, 0.65)(5, 0.65)(6, 0.65) (cid:358) (cid:358) (c) η − S Brθ ω max π α Ω (cid:358) S r (cid:84) (2, 0.45)(4, 0.55)(6, 0.65) FIG. 5: Convergence of background strain rate field S Bij ( x ) fora Burgers vortex, obtained from total strain rate field S ij ( x )using (20) for various ( n, ˜ R ) combinations, where ˜ R ≡ R/λ ν .Shown are effects of increasing ˜ R for fixed n = 6 ( top ), in-creasing n for fixed ˜ R = 0 .
65 ( middle ), and increasing n and˜ R simultaneously ( bottom ). The dashed horizontal lines fol-low from (27) and (29). In Fig. 5( b ) similar results are shown for the effect ofincreasing the order n of the expansion for the vorticityfield for fixed ˜ R = 0 .
65. It is apparent that the effect of n is somewhat smaller than for ˜ R in Fig. 5( a ). Moreover,the results suggest that the series in (20) alternates withincreasing order n . For this Ω and ˜ R , even n = 3 isseen to be sufficient to remove most of the vortex-inducedshear strain, and thus reduce S Brθ ( x ) below the horizontaldashed line. For these parameters, the S Brθ field from(20) would thus reveal alignment of the vorticity with the most extensional principal axis of the background straintensor throughout the entire field.Figure 5( c ) shows the combined effects of increasingboth ˜ R and n , in accordance with the expectation thatlarger ˜ R should require a higher order n to adequatelyrepresent the vorticity field within the spherical region.The shear strain rate field shows convergence to the cor-rect background strain field in (31). The convergence ofthe shear strain rate S Brθ ( x ) to zero in the vicinity of thevortex core is of particular importance. The systematicreduction in the peak remaining shear stress indicatesthat, even for increasingly stronger vortices or increas-ingly weaker background strain fields as measured by Ω,the resulting S Brθ ( x ) from (20) will reveal the alignmentof all the vorticity in such a structure with the most ex-tensional principal strain axis of the background strainfield. IV. VORTICITY ALIGNMENT IN TURBULENTFLOWS
Having seen in the previous Section how (20) is ableto reveal the expected alignment of vortical structureswith the most extensional eigenvector of the background strain rate in which they reside, in this Section we applyit to obtain insights into the vorticity alignment in tur-bulent flows. In particular, we examine the alignment atevery point x of the vorticity ω relative to the eigenvec-tors of the total strain rate tensor field S ij ( x ) and thoseof the background strain field data S Bij ( x ). This analysisuses data from a highly-resolved, three-dimensional, di-rect numerical simulation (DNS) of statistically station-ary, forced, homogeneous, isotropic turbulence [17, 18].The simulations correspond to a periodic cube with sidesof length of 2 π resolved by 2048 grid points. The Taylormicroscale Reynolds number R λ is 107.The DNS data were generated by a pseudospectralmethod with a spectral resolution that exceeds the stan-dard value by a factor of eight. As a result, the highestwavenumber corresponds to k max η K = 10, and the Kol-mogorov lengthscale η K = ν / / h ǫ i / is resolved withthree grid spacings. This superfine resolution makes itpossible to apply the result in (20) for relatively high or-ders n , which require accurate evaluation of high-orderderivatives of the DNS data. In Schumacher et al. [17]it was demonstrated that derivatives up to order six arestatistically converged. More details on the numericalsimulations are given in Refs. [17, 18].Figure 6 gives a representative sample of the DNS data,where the instantaneous shear component S of the totalstrain rate tensor field S ij ( x ) is shown in a typical two-dimensional intersection through the 2048 cube. Thedata can be seen to span nearly 700 η K in each direction.The 512 box at the lower left of Fig. 6 is used here toobtain initial results for alignment of the vorticity withthe eigenvectors of the background strain rate tensor.The background strain rate tensor field S Bij ( x ) is first FIG. 6: Instantaneous snapshot of total strain rate com-ponent field S ( x ) in a two-dimensional slice through ahighly-resolved three-dimensional (2048 ) DNS of homoge-neous, isotropic turbulence [17, 18]. Axes are given both ingrid coordinates ( i = 1 . . . η K . Box indicates region in which backgroundstrain rate field S Bij ( x ) is computed in Fig. 7. extracted via (20) from S ij ( x ) for n = 3 and vari-ous ( R/η K ). Higher-order evaluation of the backgroundstrain rate is not feasible, as the results in Ref. [17] showthat only spatial derivatives of the velocity field up toorder six can be accurately obtained from these high-resolution DNS data. For n = 4, the expansion in (20) in-volves seventh-order derivatives of the velocity field, andthe background strain evaluation becomes limited due tothe grid resolution. The results are shown and comparedin Fig. 7, where the shear component S of the full strainrate tensor is shown at the top, and the correspondingnonlocal (background) component S B and local compo-nent S R are shown, respectively, in the left and rightcolumns for ( R/η K ) = 2 . top row ), 3 . middle row ),and 4 . bottom row ). Consistent with the results fromthe Burgers vortex in Fig. 5, as ( R/η K ) increases themagnitude of the extracted local strain rate in the rightcolumn increases. However, for the largest ( R/η K ) = 4 . n = 3 appears to be too small to adequately rep-resent the local vorticity field. This leads to truncationerrors which are manifested as strong ripples in the back-ground and local strain fields (see panels ( f ) and ( g )).The results in Fig. 7 thus indicate that radii up to( R/η K ) = 3 . n = 3 can be used toassess alignment of the vorticity vector with the eigenvec-tors of the background strain rate field. Figure 8 showsthe probability densities of the alignment cosines for thevorticity vector with the total strain rate tensor and withthe background strain rate tensors from (20). We com-pare S ij (Fig. 8 a ) with S Bij for (
R/η K ) = 2 . , n = 3 (Fig.8 b ) and S Bij for (
R/η K ) = 3 . , n = 3 (Fig. 8 c ). The re-sults for alignment with the total strain rate tensor areessentially identical to the anomalous alignment seen in numerous other DNS studies [5, 6, 7] and experimentalstudies [8, 9, 10, 11, 12], which show the vorticity to bepredominantly aligned with the eigenvector correspond-ing to the intermediate principal strain rate. However,the results for the Burgers vortex in the previous sectionshow that such anomalous alignment with the eigenvec-tors of the total strain rate tensor is expected when thelocal vortex strength parameter Ω is sufficiently large tocause alignment switching.By comparison, the results in Fig. 8 ( b ) and ( c ) ob-tained for the alignment cosines of the vorticity vectorwith the background strain rate tensor S Bij from (20)show a significant decrease in alignment with the inter-mediate eigenvector, and an increase in alignment withthe most extensional eigenvector. While data in panel( b ) show only a slight change compared to those in ( a ),the results in panel ( c ) demonstrate that our decomposi-tion can indeed diminish the anomalous alignment signif- FIG. 7: Total strain rate component field S ( x ) ( a ), with cor-responding results from (20) for nonlocal (background) field S B ( x ) ( left ) and local field S R ( x ) ( right ) for ( R/η K ) = 2 . b, c ), ( R/η K ) = 3 . d, e ), and ( R/η K ) = 4 . f, g ), all with n = 3. (b) | cos( θ ) | (c) | cos( θ ) | (a) P ( | c o s ( θ ) | ) | cos( θ ) | e · e ω e · e ω e · e ω FIG. 8: Probability densities of alignment cosines for the vorticity with the eigenvectors of the strain rate tensor, showingresults for S ij ( a ) and for S Bij using (
R/η K ) = 2 . n = 3 ( b ) and ( R/η K ) = 3 . n = 3 ( c ). icantly. This is consistent with the results for the Burgersvortex in the previous Section, and with the hypothesisthat the alignment switching mechanism due to the localcontribution S Rij to the total strain rate tensor is the pri-mary reason for the anomalous alignment seen in earlierstudies. It is also consistent with the expected equilib-rium alignment from (1). While a more detailed study isneeded to examine possible nonequilibrium contributionsto the alignment distributions associated with eigenvec-tor rotations of the background strain field, as well asto definitively determine the R and n convergence of thebackground strain rate tensor in Fig. 7, the present find-ings support both the validity of the result in (20) forextracting the background strain rate tensor field S Bij ( x )from the total strain rate tensor field S ij ( x ), and the hy-pothesis that at least much of the anomalous alignmentof vorticity in turbulent flows is due to the differences be-tween the total and background strain rate tensors andthe resulting alignment switching noted herein. V. CONCLUDING REMARKS
We have developed a systematic and exact result in(20) that allows the local and nonlocal (background) con-tributions to the total strain rate tensor S ij at any point x in a flow to be disentangled. The approach is based ona series expansion of the vorticity field in a local sphericalneighborhood of radius R centered at the point x . Thisallows the background strain rate tensor field S Bij ( x ) tobe determined via a series of increasingly higher-orderLaplacians applied to the total strain rate tensor field S ij ( x ). For the Burgers vortex, with increasing radius R relative to the local gradient lengthscale λ ν and withincreasing order n , we demonstrated convergence of theresulting background strain tensor field to its theoreti-cal form. We also showed that even with limited R and n values, the local contribution to the total strain ratetensor field can be sufficiently removed to eliminate the anomalous alignment switching throughout the flow field.This conclusion is expected to also apply to the more re-alistic case of a non-uniformly stretched vortex where S zz = f ( z ) [16, 19, 20, 21].Consistent with the results for the Burgers vortex,when (20) was used to determine the background strainrate tensor field S Bij ( x ) in highly-resolved DNS data for aturbulent flow, the anomalous alignment seen in previousDNS and experimental studies was substantially reduced.We conclude that (20) allows the local background strainrate tensor to be determined in any flow. Furthermore,we postulate that the vorticity vector field in turbulentflows will show a substantially preferred alignment withthe most extensional principal axis of the backgroundstrain rate field, and that at least much of the anoma-lous alignment found in previous studies is simply a re-flection of the alignment switching mechanism analyzedin Section III and conjectured by numerous previous in-vestigators.Lastly, the result in (20) is based on a Taylor series ex-pansion of the vorticity within a spherical neighborhoodof radius R around any point x . Such an expansion inher-ently involves derivatives of the total strain rate tensorfield, which can lead to potential numerical limitations. Iflarger R and correspondingly higher orders n are neededto obtain accurate evaluations of background strain ratefields, then otherwise identical approaches based on al-ternative expansions may be numerically advantageous.For instance, an expansion in terms of orthonormal basisfunctions allows the coefficients to be expressed as inte-grals over the vorticity field within r ≤ R , rather than asderivatives evaluated at the center point x . (For exam-ple, wavelets have been used to test alignment betweenthe strain rate eigenvectors and the vorticity gradient intwo-dimensional turbulence [22].) This would allow a re-sult analogous to (20) that can be carried to higher orderswith less sensitivity to discretization error. The key con-clusion, however, of the present study is that it is possibleto evaluate the background strain tensor following thegeneral procedure developed herein, and that when suchmethods are applied to assess the background strain ratefields in turbulent flows they reveal a substantial increasein the expected alignment of the vorticity vector with themost extensional principal axis of the background strainrate field. Acknowledgments
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