UPTEC F11 031

Examensarbete 30 hp07 Juni 2011

Energy decay in vortices

Björn Lönn

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Energy decay in vortices

Björn Lönn

The long time energy decay of vortices for several different initial flow scenarios isinvestigated both theoretically and numerically. The theoretical analysis is based onthe energy method. Numerical calculations are done by solving the compressibleNavier-Stokes equations using a high order stable finite difference method. Thesimulations verify the theoretical conclusion that vortices decay at a slow ratecompared to other types of flows. Several Reynolds numbers and grid sizes in bothtwo and three dimensions are considered.

ISSN: 1401-5757, UPTEC F11 031Examinator: Tomas NybergÄmnesgranskare: Per LötstedtHandledare: Jan Nordström

Abstract

The long time energy decay of vortices for several different initial flowscenarios is investigated both theoretically and numerically. The theoreticalanalysis is based on the energy method. Numerical calculations are done bysolving the compressible Navier-Stokes equations using a high order stablefinite difference method. The simulations verify the theoretical conclusionthat vortices decay at a slow rate compared to other types of flows. Sev-eral Reynolds numbers and grid sizes in both two and three dimensions areconsidered.

1. Introduction

It has long been a mystery why vortices have slow energy decay. Asignificant example of vortices with slow decay, is the vortices created bythe wingtips of an airplane, see Figure 1. It is known that these vorticesalso occur at runways and remain there for a long time, making it hazardousfor other aircrafts [1]. Furthermore, vortices are important when explainingseveral flow phenomena. The lift given by flapping wings is investigated in[2]. It is concluded that effects from leading edge vortices can be used toincrease lift, which is likely important when explaining how insects fly.

Meteorological investigations often encounter vortices, for example thepresence of von Karman vortices in the trailing wind of islands, see Figure 2,can have effects on the local climate as well as the sea life [3]. Vortices are alsoneeded when modeling the general circulation of the ocean [4]. Furthermore,vortices impact the efficiency of turbine parks, as trailing vortices from theturbines causes disturbances for other turbines in the park [5]. In [6] thekinetic energy of a simulated vortex is displayed, and it is shown that theenergy decay of that vortex decrease with time.

It is shown in [7] that the vorticity decay depends on diffusion of vorticityand stretching and tilting of the vortex line. We suspect that in additionto this, eigenvalues related to the energy dissipation of the Navier-Stokesequations will emerge as a major factor when explaining why vortices decayslowly. We will explore this idea theoretically and check it by direct numericalsimulations of the compressible Navier-Stokes equations.

June 8, 2011

Figure 1: Wake vortex from an small plane made visible by colored smoke.

Figure 2: Vortices in the atmosphere created by the flow past an island.

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2. The Navier-Stokes equations

Consider the flow situation when small disturbances are imposed on aconstant background state. We investigate the decay of these small distur-bances. This problem setup also naturally allows for the linearized analysisbelow. The dependent variables are the density ρ, the velocities in each di-rection u, v, w and the temperature T . The tilde sign signifies the presenceof dimensions. We write the equations in non-dimensional form using thefree stream density ρ∞, the free stream velocity U∞ and the temperatureT∞. The shear and second viscosity coefficients µ, λ are non-dimensionalizedwith the free stream viscosity µ∞. The pressure in non-dimensional formbecomes: p = p/(ρ∞U

2∞) = ρT/(γM2

∞). Also used are

M2∞ =

U2∞

γRT∞, P r =

µ∞Cpκ∞

, Re =ρ∞U∞L

µ∞, γ =

CpCv

(1)

where κ is the coefficient of heat conduction and M , Pr, Re = 1/ε and γ arethe Mach-, Prandtl-, Reynolds number and ratio of specific heats respectively.We also need a length scale L giving the dimensionless time as t = tU∞/L.

Inserting the scalings and identifying the dimensionless factors, leads tothe dimensionless Navier-Stokes equations in two dimensions:

ρt + (ρu)x + (ρv)y = 0

ut + u˜· ∇u+

1

ρpx =

ε

ρ[[(2µ+ λ)ux + λvy]x + [µvx + µuy]y]

vt + u˜· ∇v +

1

ρpy =

ε

ρ[[(2µ+ λ)vy + λux]y + [µvx + µuy]x]

Tt + u˜· ∇T + (γ − 1)T (ux + vy) =

ε

ρ[(ϕTx)x + (ϕTy)y +M2

∞φ]

φ = γ(γ − 1)[(2µ+ λ)u2x + 2λuxvy + (2µ+ λ)v2

y + 2µuyvx + µv2x + µu2

y]

(2)

where u˜

= (u, v) and ϕ = γκPr

.

When linearizing around a constant state, the velocity gradients in φbecomes zero. The constant state is denoted with the bar sign. The linearizeddimensionless Navier-Stokes equations on matrix form are:

∂V

∂t+ A

∂V

∂x+ B

∂V

∂y+ C

∂V

∂z=ε

ρ[(D11

∂V

∂x+ D12

∂V

∂y+ D13

∂V

∂z)x+

+(D21∂V

∂x+ D22

∂V

∂y+ D23

∂V

∂z)y + (D31

∂V

∂x+ D32

∂V

∂y+ D33

∂V

∂z)z]

(3)

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where V = (ρ, u, v, w, T )T . The matrices need to be symmetric if the energymethod is to be applied. According to [8] the symmetrization can be done us-ing a similarity transformation S. Requirements for the similarity transformS was found by inspection of the Dij matrices and symmetrizing require-ments from A, B and C. We do not want the Dij matrices to be changedby the similarity transformation. Hence, one choice of similarity transformis diagonal with the second, third and fourth diagonal element equal. Thecriteria for symmetrizing A, B and C then lead to the diagonal symmetrizer:

S−1 = diag[c2

√γ, ρc, ρc, ρc,

ρ√γ(γ − 1)M4

∞] (4)

where c is the velocity of sound. The similarity transform is applied bymultiplying (3) from the left with S−1.

After applying the symmetrizer we obtain:

A =

u c/

√γ 0 0 0

c/√γ u 0 0 c

√γ−1γ

0 0 u 0 00 0 0 u 0

0 c√

γ−1γ

0 0 u

D11 =

0 0 0 0 00 2µ+ λ 0 0 00 0 µ 0 00 0 0 µ 00 0 0 0 ϕ

B =

v 0 c/

√γ 0 0

0 v 0 0 0

c/√γ 0 v 0 c

√γ−1γ

0 0 0 v 0

0 0 c√

γ−1γ

0 v

D22 =

0 0 0 0 00 µ 0 0 00 0 2µ+ λ 0 00 0 0 µ 00 0 0 0 ϕ

(5)

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C =

w 0 0 c/

√γ 0

0 w 0 0 00 0 w 0 0

c/√γ 0 0 w c

√γ−1γ

0 0 0 c√

γ−1γ

w

D33 =

0 0 0 0 00 µ 0 0 00 0 µ 0 00 0 0 2µ+ λ 00 0 0 0 ϕ

D12 = DT21 =

0 0 0 0 00 0 λ 0 00 µ 0 0 00 0 0 0 00 0 0 0 0

D13 = DT31 =

0 0 0 0 00 0 0 λ 00 0 0 0 00 µ 0 0 00 0 0 0 0

D23 = DT32 =

0 0 0 0 00 0 0 0 00 0 0 λ 00 0 µ 0 00 0 0 0 0

V =

c2√γρ

ρcuρcvρcwρ√

γ(γ−1)M4∞T

(6)

where A = S−1AS, B = S−1BS and C = S−1CS.The energy method is applied by multiplying the symmetrized dimension-

less Navier-Stokes equations with V T = (S−1V )T and integrating over thedomain Ω. Let the energy norm be defined as ||V ||2 =

∫ΩV T V dxdydz. Using

partial integration followed by Gauss’ theorem, the energy method yields:

||V ||2t +BT = −2ε

ρ

∫Ω

VxVyVz

T D11 D12 D13

D21 D22 D23

D31 D32 D33

VxVyVz

dxdydz, (7)

BT =

∮∂Ω

V T [AV − 2 ερ(D11Vx + D12Vy + D13Vz)]

V T [BV − 2 ερ(D21Vx + D22Vy + D23Vz)]

V T [CV − 2 ερ(D31Vx + D32Vy + D33Vz)]

T

n ds (8)

where ds =√dx2 + dy2 + dz2 and n = (n1, n2, n3)T is the outward-pointing

unit normal on the surface ∂Ω.In this paper we ignore the effect of farfield boundaries and consider the

Cauchy problem only. Hence we neglect the boundary term BT from nowon. Denote the block matrix consisting of the Dij matrices by D. The lefthand side now consist only of the time derivative of the norm, and if D ispositive semidefinite the right hand side dissipates energy.

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The first, sixth and 11th rows as well as the first, sixth and 11th columnsof the 15x15 matrix D have only 0 elements, thus the matrix can be reducedto a 12x12 matrix. We refer to this matrix as D.

D =

2µ+ λ 0 0 0 0 λ 0 0 0 0 λ 00 µ 0 0 µ 0 0 0 0 0 0 00 0 µ 0 0 0 0 0 µ 0 0 00 0 0 ϕ 0 0 0 0 0 0 0 00 µ 0 0 µ 0 0 0 0 0 0 0λ 0 0 0 0 2µ+ λ 0 0 0 0 λ 00 0 0 0 0 0 µ 0 0 µ 0 00 0 0 0 0 0 0 ϕ 0 0 0 00 0 µ 0 0 0 0 0 µ 0 0 00 0 0 0 0 0 µ 0 0 µ 0 0λ 0 0 0 0 λ 0 0 0 0 2µ+ λ 00 0 0 0 0 0 0 0 0 0 0 ϕ

(9)

The eigenvalues of D are obtained by solving det(D− Is) = 0. Note thatthe determinant can be expanded three times using row 4,8 and 12. Thisleaves a 9x9 determinant to be expanded further. That sub-determinantwas expanded using the symmetric properties of D. For instance, if D isexpanded using the second row, we will get two different minors which bothhave one row with only one nonzero value. If we continue by expandingeach of the minors using this row the resulting minor will be equal for both.This procedure can be repeated two times more resulting in a factorizedpolynomial multiplied by a 3x3 determinant:

0 = (ϕ− s)3[(µ− s)2 − µ2

]3 ∣∣∣∣∣∣2µ+ λ− s λ λ

λ 2µ+ λ− s λλ λ 2µ+ λ− s

∣∣∣∣∣∣ . (10)

By expanding the determinant in (10) we get the final equation

det(D − Is) = (ϕ− s)3[(µ− s)2 − µ2)

]3(2µ− s)2(2µ+ 3λ− s) = 0. (11)

The eigenvalues are ϕ, ϕ, ϕ, 2µ, 2µ, 2µ, 0, 0, 0, 2µ, 2µ, 2µ + 3λ. To see howthe velocity gradients depend on the eigenvalues, the eigenvectors must be

8

calculated. The orthonormal matrix with the eigenvectors as columns is:

Q =

0 0 0 0 0 0 0 0 0√

16

1√2

1√3

0 0 0 1√2

0 0 1√2

0 0 0 0 0

0 0 0 0 1√2

0 0 1√2

0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 00 0 0 1√

20 0 −1√

20 0 0 0 0

0 0 0 0 0 0 0 0 0 −√

46

0 1√3

0 0 0 0 0 1√2

0 0 1√2

0 0 0

0 1 0 0 0 0 0 0 0 0 0 00 0 0 0 1√

20 0 −1√

20 0 0 0

0 0 0 0 0 1√2

0 0 −1√2

0 0 0

0 0 0 0 0 0 0 0 0√

16− 1√

21√3

0 0 1 0 0 0 0 0 0 0 0 0

. (12)

The matrix D can now be written as D = QΛQT where

Λ = diag(ϕ, ϕ, ϕ, 2µ, 2µ, 2µ, 0, 0, 0, 2µ, 2µ, 2µ+ 3λ). (13)

Inserting this expression of D into (7) leads to:

||V ||2t = −2ε

ρ

∫D

QT

VxVyVz

T Λ

QT

VxVyVz

dxdydz , (14)

where

QT

VxVyVz

=

ρ√γ(γ−1)M4

∞[∇T ]

1√2ρc[vx + uy]

1√2ρc[wx + uz]

1√2ρc[wy + vz]

1√2ρc[vx − uy]

1√2ρc[wx − uz]

1√2ρc[wy − vz]√

23ρc[1

2ux − vy + 1

2wz]

1√2ρc[ux − wz]

1√3ρc[ux + vy + wz]

. (15)

9

Equation (14) and (15) show the dependence between eigenvalues andvelocity gradients. The temperature gradients are multiplied by the eigen-value ϕ. The following three components represent the rotation of the flowvector field in the xy-, xz- and yz planes and are each multiplied by 2µ. Thevorticity in each plane is multiplied by a zero eigenvalue. The last term isthe divergence and it is multiplied by 2µ + 3λ. It is unclear how the tworemaining terms should be interpreted. However, combined with the diver-gence term they become (u2

x + v2y + w2

z). This suggest that in order to getslow decay it is not enough to have small divergence, unless each componentof the divergence is also small. The dissipation is thus determined by thetemperature gradients, rotation, vorticity and divergence of the flow.

The decay is proportional to the gradients and a flow scenario with largegradients should decay fast, compared to a scenario with small gradients.Note that since the vorticity terms are multiplied by the zero eigenvalues thecontribution to the decay from these terms will always be zero. Hence vorticesdecay slowly because the energy of the flow is focused into vorticity, whichmakes all other terms close to zero, if the temperature terms are ignored.

3. Numerical Calculations

The computations are done using a stable well tested high order finitedifference scheme for the compressible Navier-Stokes equations as describedin [9-12]. The 4th order scheme with far field boundaries sufficiently far fromthe actual flow structure was used.

3.1. Grid generation

The grids used for calculations in two dimensions are each divided into16 blocks. All blocks have an equal number of grid points. The grid used, forthe displayed computations, is a square with side length 20 and 257 pointseach in both the x and y directions. Hence the mesh has ∆x = ∆y = 0.078.For calculations in three dimensions the grid is a cube with side length 20and it has 129 points in each direction giving ∆x = ∆y = ∆z = 0.155. Bycomputing with successively coarser grids we determined that all the initialdata considered, were properly resolved.

3.2. Initial conditions in two dimensions

To verify the analytical results, long time simulations with three differentinitial conditions have been investigated. The Reynolds numbers considered

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are Re = 10 and Re = 100. We stress here that the zero eigenvalues arepresent for all Reynolds numbers and therefore the size of the Reynoldsnumber is of no importance in this investigation. We have compared theenergy decay of a vortex, with the energy decay of white noise, and a parallelflow case. The initial values for all these two dimensional calculations are:the Mach number M=0.005, the radius r=1, the density and temperature areρ∞ = 1, T∞ = 1.

The vortex is a so called wake vortex also used in [6]. The initial velocityfield is shown in Figure 3 and defined by

uij = − ε · yijx2ij + y2

ij + r2vij =

ε · xijx2ij + y2

ij + r2 (16)

where xij, yij are the distances from the center of the vortex in each direction,r is the distance from the center to the strongest velocity of the vortex andε = 2rM .

Figure 3: The initial field of a wake vortex computation with vortex radiusr=1. The velocities decrease fast outside r=1.

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The velocity field of the white noise initial condition is shown inFigure 4 and defined by

uij =ε · rand

x2ij + y2

ij + r2vij =

ε · randx2ij + y2

ij + r2rand ∈ [−1, 1] (17)

where ε is the same as in (16). The velocities are randomized using randomnumbers from the Fortran 90 random subroutine and damped to zero usingthe same function as the vortex condition.

Figure 4: The initial field of the white noise. The velocities are strongestwithin the radius r=1.

The velocity field of the initial condition with parallel flow is defined as

uij =ε

x2ij + y2

ij + r2vij =

ε

x2ij + y2

ij + r2 (18)

where ε is the same as in (16). Initially the velocities are all directed parallelto the line y = x and decreasing with the distance from the center as shownin Figure 5.

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Figure 5: The initial field of the parallel velocity initial condition. Thevelocities are strongest within the radius r=1.

The time evolution of the wake vortex can be seen in Figure 6 where thevortex is displayed at four different times. The kinetic energy of the vortexspread evenly to the area surrounding it. This can be seen by observing howthe density and velocities change with time. At t=4000 the vortex seemsvery weak. But as we will see later, this is an illusion.

The time evolution of the white noise computation is displayed in Fig-ure 7. The white noise decays quickly and has almost vanished at t=1000.Initially a wave moves away from the center of the noise and out of the com-putational domain. A closer look at what remain of the white noise at t=500and t=1000 reveal several vorticity like flow structures.

The time evolution of the parallel flow velocity initial condition (18) isdisplayed in Figure 8. With the parallel velocity initial condition the flowdevelops into two counter rotating vortices. Just as the vorticity like struc-tures in all other computations, these vortices remain for a long time andseem to decay slowly.

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(a) t=0

(b) t=50

Figure 6: Flow evolution of the wake vortex computation. The radius isdisplayed by the low density core indicated with dark blue in (a).

14

(c) t=1000

(d) t=4000

Figure 6: Flow evolution of the wake vortex computation. With time thecore and the velocities weaken (c)-(d).

15

(a) t=0

(b) t=6

Figure 7: Flow evolution of the white noise initial condition.

16

(c) t=500

(d) t=500 zoomed

Figure 7: Flow evolution of the white noise initial condition.

17

(e) t=1000

(f) t=1000 zoomed

Figure 7: Flow evolution of the white noise initial condition.

18

(a) t=0

(b) t=6

Figure 8: Time evolution of parallel flow computation. Initially a wave movesaway from the center, and it leaves the computational domain at t=10.

19

(c) t=100

(d) t=600

Figure 8: Time evolution of parallel flow computation.

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3.3. Energy decay in two dimensions

The kinetic energy of the global system E = ρu2+v2+w2

2as a function of

time, normalized by the initial kinetic energy, is show in Figure 9.

Figure 9: Normalized kinetic energy decay for vortex, parallel and whitenoise initial conditions.

The energy decay of white noise is clearly faster compared to the energydecay of the wake vortex and the parallel flow case. The fast decay is pre-dicted by the theory since white noise has large gradients initially comparedto the other initial conditions. Note that once the white noise has developedinto a flow consisting mainly of vorticity, see Figure 7, the decay stops. Theparallel energy decay is fast initially when the velocity gradients are strongand decreases as the flow becomes dominated by two counter rotating vor-tices. It is reasonable that the strongest velocities decay faster since theyalso produce larger gradients, just like in the other computations. Note thatonce the vortices have formed, the decay is similar to the wake vortex de-cay. The energy of the vortex decrease slowly with time which agrees withthe results obtained in [6]. Despite the fact that the wake vortex flow withRe=100 looks weaker at t=4000 more than 90% of the kinetic energy in theglobal system remains. Computing with a lower/higher Reynolds numberconsistently resulted in more/less energy decay.

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3.4. Computations in three dimensions

In three dimensions, computations for a wake vortex, white noise andparallel flow have been made. The initial velocities are defined as:

V ortex : uijk =−0.1 · ε · yijkf(x, y, z)

, vijk =0.1 · ε · xijkf(x, y, z)

, wijk = 0 (19)

Noise : uijk =0.1 · ε · random

f(x, y, z), vijk =

0.1 · ε · randomf(x, y, z)

, wijk = 0 (20)

Parallel : uijk =0.1 · ε

f(x, y, z), vijk =

0.1 · εf(x, y, z)

, wijk = 0 (21)

were f(x, y, z) = x2ijk + y2

ijk + z2ijk + r2 and the same parameter values as in

two dimensions were used. Images of a three dimensional vortex followed byan engineered vorticity case are confined to Appendix A. The energy of theglobal system as a function of time is shown in Figure 10. The focus of the

Figure 10: Kinetic energy as a function of time for three dimensional vortex,white noise and parallel flow initial conditions.

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computations in three dimensions is to verify the relevance of the computa-tions in two dimensions. Far field boundary conditions are used on all outerboundaries, except in the z-direction. Since the vortex has its vortex lineparallel to the z-direction, periodic boundary conditions are used in this di-rection. In all three computations the energy decay is similar to and hencevalidate the energy decay obtained from the two dimensional computations.

4. Conclusions

By deriving an equation for the energy decay in terms of gradients anda dissipation matrix it was shown that the vorticity components were multi-plied by zero eigenvalues. This indicates that flows with most of the energyin ”vorticity” form should decay slowly.

Direct numerical simulations in two and three dimensions verified thatthe energy decay is slow for vortices compared to white noise and parallelflow. This indicates that once the flow is dominated by vorticity structuresit decays slowly. Furthermore, an increased Reynolds number decreases theenergy decay as it should.

The existence of the zero eigenvalues multiplying the vorticity componentof the flow is probably one of the main reasons for the slow decay of energyin vortex like structures in fluid mechanics.

AcknowledgmentsThe data used in this effort were acquired as part of the activities of

NASA’s Science Mission Directorate, and are archived and distributed bythe Goddard Earth Sciences (GES) Data and Information Services Center(DISC).

AppendixA. Three dimensional flow cases

In Figure A.11 a three dimensional wake vortex is displayed. The initialvelocities are defined as:

uij =−ε · yijk

x2ijk + y2

ijk + r2, vijk =

ε · xijkx2ijk + y2

ijk + r2, wijk = 0. (A.1)

The geometry and rotation of the vortex is depicted by iso-surfaces of theenergy and velocities. The vortex is cylinder shaped and the iso-surfaces ofthe velocities display the counterclockwise rotation.

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(a) Iso-surface of the energy

(b) Iso-surfaces of x-velocity (c) Iso-surfaces of y-velocity

Figure A.11: Wake vortex in three dimensions. In the velocity figures bluecorresponds to negative-, green corresponds to zero- and yellow correspondsto positive velocity.

A flow case manufactured from the theoretical conclusions in three di-mensions is shown in Figure A.12. Iso-surfaces of the energy visualize thegeometry of the flow which is similar, yet different from the wake vortex.The vorticity of the flow is displayed by iso-surfaces of the velocities. Thevelocities are defined as:

uij =ε · (−yijk − zijk)

f(x, y, z), vijk =

ε · (xijk − zijk)f(x, y, z)

, wijk =ε · (xijk + yijk)

f(x, y, z)(A.2)

were f(x, y, z) = x2ijk + y2

ijk + z2ijk + r2. The goal was to manufacture a case

24

with slow energy decay using Equation (15). Hence the initial velocities areconstructed so that the vorticity in the xy-, xz-, and yz-planes is the same,vx − uy = uz − wx = vz − wy = K were K is a constant. This can be seen inthe velocity figures of Figure A.12.

(a) Iso-surfaces of the energy (b) Iso-surfaces of x-velocity

(c) Iso-surfaces of y-velocity (d) Iso-surfaces of z-velocity

Figure A.12: Flow with vorticity in the xy-, xz- and yz-planes simultaneously.Blue corresponds to low energy and yellow corresponds to high energy. Inthe velocity figures blue corresponds to negative-, green corresponds to zero-and yellow corresponds to positive velocity.

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