some novel features of three dimensional ...rupak mukherjee rajaraman ganesh abhijit sen institute...

1/21

Motivation G-MHD3D Nonlin Coherent Oscillation Recurrence Dynamo in 3D MHD Summary Extra

Some Novel Features of Three DimensionalMagnetoHydroDynamic Plasma

G-MHD3D: DNS Code for 3D MHD Plasma Modelling

Rupak Mukherjee Rajaraman Ganesh Abhijit Sen

Institute for Plasma Research, HBNI, Gandhinagar, Gujarat, India

[email protected]

[email protected]

[email protected]

[email protected]

Princeton Plasma Physics Laboratory12 November, 2018

Rupak Mukherjee IPR, India

Nonlinear Coherent Oscillation, Recurrence and Dynamo in 3D MHD

2/21


Aim of the Talk

Direct Numerical Simulation (DNS) study of 3D Single fluidMagnetoHydroDynamic equations have been carried out to explore

1 Nonlinear Coherent Oscillation - a energy oscillation betweenkinetic and magnetic modes. RM, R Ganesh, Abhijit Sen; arXiv:1811.00744

2 “Recurrence Phenomena” - a periodic reconstruction of initialflow of fluid & magnetic variables. RM, R Ganesh, Abhijit Sen; arXiv:1811.00754

3 “Dynamo Effect” - Mean and intermediate scale magneticfield generation from driven chaotic flows. [Ongoing]

1 Finite mode / Galerkin representation confirms nonlinearinteraction of few modes.

2 Rayleigh Quotient determines criteria of recurrence. [KeyParameter: Initial Condition]

3 Parameter set for fast dynamo is explored [Key Parameters:Forcing amplitude (~f0) & Driving scale (kf )]



3/21


Governing Equations in three dimensions

Single Fluid 3D-MHD equations evaluated by G-MHD3D∂ρ

∂t+ ~∇ · (ρ~u) = 0

∂(ρ~u)

∂t+ ~∇ ·

[ρ~u ⊗ ~u + Ptot

¯̄I− 1

4π~B ⊗ ~B−2νρ¯̄S

]= ρ~F

∂E

∂t+ ~∇ ·

[(E + Ptot) ~u−

1

4π~u ·(~B ⊗ ~B

)][−2νρ~u · ¯̄S − η

4π~B ×

(~∇× ~B

)]= 0

∂ ~B

∂t+ ~∇ ·

(~u ⊗ ~B − ~B ⊗ ~u

)= η∇2 ~B

Sij =1

2(∂iuj + ∂jui )−

1

3δij ~∇ · ~u

Ptot = Pth +1

8π|~B|2; Pth = C 2

s ρ



4/21


Scope of the code

Evaluates single fluid 3-D MagnetoHydroDynamic equations.

Pseudo-spectral technique is more accurate and faster thanstandard finite difference methods. [FFTW & cuFFT library]

Can handle effects arising due to weak compressibility.

3D Isosurface reconstruction is developed using Mayavi.

Diagnostics viz. Energy spectra, Particle & Field Line tracer(Cloud-In-Cell & Velocity Verlet scheme), Poincare Sectionare developed. [In collaboration with Vinod Saini, IPR]

GPU code optimisation is achieved with OpenACC and CUDAparallelisation. [With Naga Vydyanathan, NVIDIA, India]

Multi-GPU parallalisation (for high resolution runs) withNV-Link is in progress to run on NVIDIA-DGX station.[Incollaboration with Naga Vydyanathan, NVIDIA, India]

RM, R Ganesh, V Saini, U Maurya, N Vydyanathan, B Sharma, Accepted in conf proceed of HiPC Workshop, 2018



5/21


Motivation

Nonlinear Coherent Oscillation

Within the premise of Single fluid MHD, energy can cascadethrough both kinetic and magnetic channels simultaneously.

With weak resistivity, MHD model predicts -1). irreversible conversion of magnetic energy into fluid kineticenergy (i.e. reconnection).2). conversion of kinetic energy into mean large scalemagnetic field (i.e. dynamo).

Question: For a given fluid type and magnetic field strength,are there fluid flow profiles which do neither?

Answer: Yes. For a wide range of initial flow speeds or AlfvenMach number it is shown that the coherent nonlinearoscillation persist.



6/21


Nonlinear coherent oscillation in two spatial dimensions at Alfven resonance (Cs = VA).

Orszag-Tang Flow

ux = −A sin(k0y)

uy = +A sin(k0x)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

Sh

ifte

d K

ine

tic,

Ma

gn

etic a

nd

To

tal E

ne

rgy

Time

Kinetic Energy

Magnetic Energy

Total Energy

Cat’s Eye Flow

ux = + sin(k0x) cos(k0y)− A cos(k0x) sin(k0y)

uy = − cos(k0x) sin(k0y) + A sin(k0x) cos(k0y)

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 2 4 6 8 10

Shifte

d Kine

tic, M

agne

tic an

d Tota

l Ene

rgy

Time

Kinetic Energy

Magnetic Energy

Total Energy

u0 = Lt0

, VA = B04π√ρ0

, Ms = u0Cs

, MA = u0VA

, Re = ρ0u0Lν , Rm = Lu0

η .

• Parameters: N = 1282, U0 = 1, A = 1(OT); 0.5(CE), k0 = 1, Ms = 0.01, MA = 1 and Re = Rm = 10−4Rupak Mukherjee IPR, India


7/21


In three spatial dimensions

Nonlinear Coherent Oscillation at Alfven Resonance(Sound speed = Alfven speed) in 3D

Kinetic & magnetic energy for Roberts & TG flows.

Roberts flow:

ux (0) = U0 sin(kz)

uy (0) = U0 sin(kx)

uz (0) = U0 sin(ky)

Taylor-Green (TG) flow:

ux (0) = A U0 [cos(kx) sin(ky) cos(kz)]

uy (0) = −A U0 [sin(kx) cos(ky) cos(kz)]

uz (0) = 0

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0 20 40 60 80 100

Sh

ifte

d E

ne

rgy

Time

Kinetic Energy

Magnetic Energy

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

1 10

E(k

)

k

time = 31.2

time = 46.8

time = 62.4

time = 78.0

time = 93.6

time = 109.2

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

0 20 40 60 80 100

Sh

ifte

d E

ne

rgy

Time

Kinetic Energy

Magnetic Energy

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1 10

E(k

)

k

time = 31.2

time = 46.8

time = 62.4

time = 78.0

time = 93.6

time = 109.2

N L dt ρ0 U0 Re Rm Ms MA A = B = C k0

643 2π3 10−5 1 0.1 450 450 0.1 1 1 1



8/21


Observations

2D Orszag-Tang Flow withExternal Forcing

~f = α

[−A sin(kf y)+A sin(kf x)

], α = 0.1

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10

Shi

fted

Kin

etic

, Mag

netic

and

Tot

al E

nerg

y

Time

Kinetic Energy

Magnetic Energy

Total Energy

3D ABC Flow with different MA

ux = U0[A sin(k0z) + C cos(k0y)]uy = U0[B sin(k0x) + A cos(k0z)]uz = U0[C sin(k0y) + B cos(k0x)]

Frequency of energy exchangescales linearly with MA.

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

0 5 10 15 20

Sh

ifte

d K

ine

tic E

ne

rgy

Time

MA = 0.1

MA = 0.2

MA = 0.3

MA = 0.4

MA = 0.5

With external forcing similar to initial flow the plasma acts asa forced-relaxed system both in two and three dimensions.



9/21


Galerkin / Finite mode representation in two spatial dimensions

Galerkin representation of the field variables (stream function ψ; vector potential A)

ψ(x , y) = ψ0 sin kx + e iky (ψ1 + ψ3 cos kx) + e−iky (ψ∗1 + ψ∗3 cos kx)

A(x , y) = A0 sin kx + e iky (A1 + A3 cos kx) + e−iky (A∗1 + A∗3 cos kx)

dψ0

dt= ik2(ψ∗3ψ1 − ψ3ψ

∗1)

dψ1

dt=

i

2k2(ψ0ψ3 + A0A3)

dψ3

dt= 0

dA0

dt= ik2(A3ψ

∗1 − A∗3ψ1 + A1ψ

∗3 − A∗1ψ3)

dA1

dt=

i

2k2(A0ψ3 − A3ψ0)

dA3

dt= ik2(A0ψ1 − A1ψ0)

-1

-0.5

0

0.5

1

0 5 10 15 20

[ψ1(t

) -

ψ1(0

)];

[A

1(t

) -

A1(0

)]

Time (t)

ψ1(t) - ψ1(0)

A1(t) - A1(0)

Initial Conditionψ0 = 1, ψ1 = 0 = ψ3

A0 = A1 = A3 = 1RM, R Ganesh, Abhijit Sen, arXiv:1811.00744



10/21


Key results of Nonlinear Coherent Oscillation

Summary of Nonlinear Coherent Oscillation

For decaying flows at Alfven resonance (Cs = VA), an initialuniform magnetic field profile leads to nonlinear coherentoscillation between kinetic and magnetic modes.

For externally driven flows, the nearly idealmagnetohydrodynamic plasma acts as a forced-relaxed system.

The oscillation can be captured through a finite modeexpansion of the MHD equations indicating the energycontent is primarily in the large scales of the system.

As the Alfven Mach number is increased further, a tendencyto mean field dynamo is seen.



11/21


Introduction to Recurrence

RECURRENCE in 3D MHD

Reconstruction of initial condition in statistically large degreesof freedom systems - counterintuitive to laws ofthermodynamics and entropy - trapping in phase space.

Recurrence was first observed in 1D FPU system.

In 2D hydrodynamic systems, reconstruction of initial flowfield was numerically observed by Yen & Ferguson andexplained by A Thyagaraja.

However, generalising the analytical argument of recurrence inhigher dimensional systems is quite challenging.

We observe recurrence in single fluid 3D MHD system.

We numerically extrapolate the previous analytical argumentand find reasonable agreement with our DNS data.



12/21


Recurrence - TG (X) flow

Kinetic Isosurfaces

Magnetic Isosurfaces



13/21


No Recurrence - ABC (7) flow

Kinetic Isosurfaces

Magnetic Isosurfaces



14/21


TG Flow & Roberts Flow

Recurrence for TG & Roberts flow



15/21


ABC Flow & Cats Eye flow

No recurrence for ABC & Cats eye flow



16/21


WHY DOES IT RECUR?

Rayleigh quotient

• Rayleigh quotient [Q(t)] measures the number of effective‘active degrees of freedom’. [Thyagaraja, Phys. Fluids, 22 (11), 2093; Thyagaraja, Phys.

Fluids, 24 (11), 1973]

• Q(t) =

∫V

[(~∇×~u

)2+ 1

2

(~∇×~B

)2]dV∫

V

(|~u|2+ 1

2|~B|2

)dV

=

∑k

k2|ck |2

∑k|ck |2

where, ~u & ~B are expanded in a Fourier series.

If Q(t) is bounded ⇒Recurrence can happen.

For TG & Roberts flow,Q(t) is bounded.

For ABC & Cats eyeflow Q(t) increaseswithout bound.

RM, R Ganesh, Abhijit Sen,

arXiv:1811.00754

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80 100

Q(t

)

Time (t)

ABC Flow

TG Flow



17/21


Summary of Recurrence

Summary of Recurrence

Ideally very low probability of trapping in phase space in highdimensional systems (e.g. 3D MHD systems).

Recurrence is observed for flows involving few number ofactive degrees of freedom. [Birkhoff, Dynamical Systems,1927, Chapter 7]

Generating the flows in experimental systems is achievable -hence can be tested in laboratory experiments.

Recurrence can be helpful to make short time forecasts oncetypical initial profiles are experimentally obtained.

Dissipative regularisation of 3D MHD discards Hamiltoniandescription leading to weak deviation from initial profiles.

Conservative regularisation of 3D MHD [Thyagaraja, PhysPlasmas 17, 032503 (2010)] may offer better recurrence.



18/21


Introduction to Driven “self consistent” Dynamos

Introduction to Driven “self consistent” Dynamos

Dynamo ⇒ Growth of magnetic energy at the cost of kineticenergy. [Mean field, Stretch-Twist-Fold (short scale) etc.][Parker, ApJ, 122, 293 (1955), Cowling, MNRAS, 94, 39,(1933), Hotta, Science, 351, 6280 (2016)]

Work is in progress ⇒ Preliminary “fast” dynamo results.

Results from 643 resolution runs. Higher resolution runs willbe performed in multi-GPU code.

It was shown by Alexakis, ABC field provides fastest kinematicdynamos [Phys Rev E, 84, 026321 (2011)].

We have tried to find optimised parameter set to obtain fastSTF dynamos using ABC field.



19/21


‘Self-consistent’ Driven Dynamo / Driven Dynamo with back-reaction at Pm = 1 and high MA

Search for “fast” dynamo with externally driven ABC flow

‘Self-consistent’ dynamo saturatesunlike kinematic dynamo.

Linear growth rate (γ) increases withmagnitude of forcing & kf .

Independent ofinitial condition.

Multistep growth ofmagnetic energy.

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

100

0 20 40 60 80 100

Energ

y

Time

A = 0.1; Kinetic Energy

A = 0.1; Magnetic Energy


A = 0.2; Magnetic Energy


A = 0.3; Magnetic Energy 1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

100

0 10 20 30 40 50

Energ

y

Time

kf = 1; Kinetic

kf = 1; Magnetic

kf = 2; Kinetic

kf = 2; Magnetic

kf = 4; Kinetic

kf = 4; Magnetic

kf = 8; Kinetic

kf = 8; Magnetic

kf = 16; Kinetic

kf = 16; Magnetic

Figure: Saturation of self-consistent Dynamo action for different magnitude & wave-number of external forcing

with parameters N = 643 L = 2π3, δt = 10−4, ρ0 = 1, U0 = 0.1, MA = 1000, Ms = 0.1, Re = Rm = 450,A = B = C = 0.1, 0.2, 0.3 and kf = 1, 2, 4, 8, 16 using MHD3D.



20/21


Preliminary results on self-consistent externally driven STF dynamo from GMHD3D

Kinetic and Magnetic Energy Spectra indicating STF dynamo

• Shell averaged kinetic energy spectra saturates with slope −5/3.• Shell averaged magnetic energy spectra saturates with slope 0.7.

1e-18

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

100

1 10

E(k

)

k

time = 0

time = 10

time = 20

time = 30

time = 40

time = 50

time = 100

time = 150

time = 200

x-5/3

1e-25

1e-20

1e-15

1e-10

1e-05

1

1 10

B(k

)

k

time = 0

time = 10

time = 20

time = 30

time = 40

time = 50

time = 100

time = 150

time = 200

x0.7

RM, R Ganesh, Abhijit Sen, Under preparation

Preliminary energy spectra shows large and intermediatescales in magnetic variables are energetically dominant.

High resolution runs are needed to explore short scales.⇒ Need for Multi-GPU G-MHD3D code.



21/21


Summary & Discussions

Summary & Discussions

Coherent nonlinear oscillation between fluid and magneticenergies is numerically observed withing the framework ofsingle fluid MagnetoHydroDynamics.

Reconstruction of initial flow data is found to occur for 3DMHD system of equations. ⇒ Recurrence

Fast dynamos are observed with chaotic flow lines at Prandtlnumber unity.

Acknowledgement

A Thyagaraja, Culham Labs, UK

Vinod Saini, Udaya Maurya, IPR, India

Nagavijayalakshmi Vydyanathan, NVIDIA, India

Thank YouRupak Mukherjee IPR, India


22/21


Taylor-Woltjer / Qin

Motivation

A steady state for near ideal plasma was derived by L. Woltjerin 1954 by extremising the free energy constructed out ofmagnetic helicity (Hm =

∫~A · ~BdV ) and magnetic energy

(Em =∫

B2dV and ~B · n̂ = 0).

This particularly yielded ~∇× ~B = α(x)~B subject to~B · ~∇α = 0 on the “walls”. α = α(x) represents the rigidity oflarge number of volume (Hm =

∫V~A · ~BdV ).

J. B. Taylor considered a particular limit where, a weakdissipation would break all the local helicity constants exceptthe one considered over the vessel with conducting surfaces,resulting in α(x) = α0 = constant and thereby, ~∇× ~B = α0

~B.

Thus it was known that large scales do not participate in theplasma relaxation until H. Qin et al [PRL, 109, 235001(2012)] who has given arbitrary scale relaxation model.



23/21



Preliminary Numerical Experiment: Taylor - Woltjer / Qin

We numerically evolve a three dimensional MHD plasma froma Beltrami class of solution in a region bounded by conductingwalls (~B · n̂ = 0 = ~u · n̂) and “suddenly” allow to expand theplasma and fill the new volume.

The expansion mediates via reconnection of magnetic fieldlines thereby flow of kin and mag energy between scales.

We measure the scales involved during this relaxation of theplasma and from our numerical tools attempt to identify amodel of plasma relaxation.

References

Taylor, PRL, 33, 1139 (1974), Woltjer, PNAS, 44, 489 (1958)

H Qin et al, PRL, 109, 235001 (2012)



24/21



3e-06

4e-06

5e-06

6e-06

7e-06

8e-06

9e-06

1e-05

1.1e-05

1.2e-05

0 20 40 60 80 100 8e-09

9e-09

1e-08

1.1e-08

1.2e-08

1.3e-08

1.4e-08

Tot

al E

nerg

y

Mag

netic

Hel

icity

Time

Re = 1000, Rm = 5000; k = 1; MA = 10; U0 = 0.1; A = B = C = 0.1

Energy

Magnetic Helicity

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1 10 100

Kin

etic E

nerg

y [E

(k)]

Wavenumber [k]

Re = 1000, Rm = 5000; k = 1; MA = 10; U0 = 0.1; A = B = C = 0.1

Time = 0

Time = 10

Time = 20

Time = 40

Time = 60

Time = 80

Time = 100 1e-15

1e-14

1e-13

1e-12

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

1 10 100

Mag

etic

Ene

rgy

[B(k

)]

Wavenumber [k]

Re = 1000, Rm = 5000; k = 1; MA = 10; U0 = 0.1; A = B = C = 0.1

Time = 0

Time = 10

Time = 20

Time = 40

Time = 60

Time = 80

Time = 100

3e-06

4e-06

5e-06

6e-06

7e-06

8e-06

9e-06

1e-05

1.1e-05

1.2e-05

0 20 40 60 80 100 8e-11

9e-11

1e-10

1.1e-10

1.2e-10

1.3e-10

1.4e-10

Tot

al E

nerg

y

Mag

netic

Hel

icity

Time

Re = 1000, Rm = 5000; k = 1; MA = 100; U0 = 0.1; A = B = C = 0.1

Energy

Magnetic Helicity

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1 10 100

Kin

etic E

nerg

y [E

(k)]

Wavenumber [k]

Re = 1000, Rm = 5000; k = 1; MA = 100; U0 = 0.1; A = B = C = 0.1

Time = 0

Time = 10

Time = 20

Time = 40

Time = 60

Time = 80

Time = 100 1e-17

1e-16

1e-15

1e-14

1e-13

1e-12

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1 10 100

Mag

etic

Ene

rgy

[B(k

)]

Wavenumber [k]

Re = 1000, Rm = 5000; k = 1; MA = 100; U0 = 0.1; A = B = C = 0.1

Time = 0

Time = 10

Time = 20

Time = 40

Time = 60

Time = 80

Time = 100

2e-07

2.2e-06

4.2e-06

6.2e-06

8.2e-06

1.02e-05

1.22e-05

0 20 40 60 80 100 0

2.01e-11

4.01e-11

6.01e-11

8.01e-11

1.001e-10

1.201e-10

1.401e-10

Tot

al E

nerg

y

Mag

netic

Hel

icity

Time

Re = 1000, Rm = 5000; k = 4; MA = 100; U0 = 0.1; A = B = C = 0.1

Energy

Magnetic Helicity

1e-18

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1 10 100

Kin

etic E

nerg

y [E

(k)]

Wavenumber [k]

Re = 1000, Rm = 5000; k = 4; MA = 100; U0 = 0.1; A = B = C = 0.1

Time = 0

Time = 10

Time = 20

Time = 40

Time = 60

Time = 80

Time = 100 1e-16

1e-15

1e-14

1e-13

1e-12

1e-11

1e-10

1e-09

1e-08

1e-07

1 10 100M

aget

ic E

nerg

y [B

(k)]

Wavenumber [k]

Re = 1000, Rm = 5000; k = 4; MA = 100; U0 = 0.1; A = B = C = 0.1

Time = 0

Time = 10

Time = 20

Time = 40

Time = 60

Time = 80

Time = 100



25/21



Preliminary Results

1 The parameters chosen are such that magnetic helicity (Hm)remains constant while HG decays allowing a Taylor-likesituation.

2 For wavenumber k = 1, for all values of the parametersstudies, power in magnetic energy spectra is seen to increasewhile the power in kinetic energy spectra decreases with time.

3 For k = 4, 8, the behavior is opposite for ~B spectra.

4 As plasma expands from time t = 0 with k = 1 from a smallvolume to fill up the simulation volume, in general while thewhole spectra is seen to contribute, mode numbers k ∼ 10 areseen to participate in the relaxation process more dominantly.

5 For k = 4, 8, the relaxation process is dominantly controlledby k>10 modes.



26/21


Derivation of single Fluid MHD Equation

∂ρ

∂t+ ~∇ · (ρ~u) = 0

~j =1

4π~∇× ~B

~E = −~u × ~B

c+ η~j

∂~u

∂t+ (~u · ~∇)~u =

µ

ρ∇2~u − 1

ρ~∇P +

1

ρ(~j × ~B)

∂ ~B

∂t= −c ~∇× ~E

~∇ · ~B = 0d

dt

(P

ργ

)= 0



27/21



~∇× ~E = −1

c~∇× (~u × ~B) + ~∇× (η~j)

= −1

c~∇× (~u × ~B) +

η

4π~∇× (~∇× ~B)

= −1

c~∇× (~u × ~B) +

η

4π∇2 ~B [∵ ~∇ · ~B = 0]

~∇× (~u × ~B) = (~B · ~∇)~u − (~u · ~∇)~B + ~u(~∇ · ~B)− ~B(~∇ · ~u)

Assumptions : P = γρKT ⇒ d

dt

(P

ργ

)≡ 0 identically

η = 0

Definition : MA =U0

VA=| ~U0|√

4πρ

|~B|



28/21



Vector Identities : (~A× ~B)× ~C = − ~C × (~A× ~B)

~A× (~∇× ~B) + ~B × (~∇× ~A) =

~∇(~A · ~B)− (~A · ~∇)~B − (~B · ~∇)~A

⇒ ~B × (~∇× ~B) + ~B × (~∇× ~B) = ~∇(~B · ~B)− (~B · ~∇)~B − (~B · ~∇)~B

⇒ 2~B × (~∇× ~B) = ~∇(~B2)− 2(~B · ~∇)~B

⇒ ~B × (~∇× ~B) =1

2~∇(~B2)− (~B · ~∇)~B

⇒ (~∇× ~B)× ~B = (~B · ~∇)~B − 1

2~∇(~B2)



29/21


Benchmark of MHD2D: Hydrodynamic (KH)

Initial Condition: Kelvin-Helmholtz flow

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6

ω

X

w0 = 2; d = 3 pi/ 128

- w0/cosh((x-pi/2)/d)**2 + w0/(cosh((x-3*pi/2)/d))**2

0

1

2

3

4

5

6

3.05 3.1 3.15 3.2Y

X

Initial Velocity

"Initial_Velocity.dat" u 1:2:($4)/3:($5)/3

Figure: Vorticity Velocity

• ν = 0.0001



30/21



KH Instability with Tracer Particles (C-I-C & VV scheme)

Figure: KH Instability for Incompressible Flow



31/21



Benchmarking with Analytical result

Analytical Growth Rate for a Broken Jet

Drazin, P. (1961). Journal of Fluid Mechanics, 10: 571-583.

γ =kxU0

3

√3− 2kxRE− 2

{(kxRE

)2

+ 2√

3kxRE

} 12

RE =

U0d

ν

d = Shearing Length



32/21



Evaluation of Growth rate of KH Instability

Numerical Growth Rate

1e-05

0.0001

0.001

0.01

0.1

1

10

0 2 4 6 8 10

KE

y

Time

2*Γ = 3.0

"k_1/Energy.dat" u 1:3"k_1/Energy.dat" u ($1)+0.0:(exp(3.0*($1)))*1e-05

1e-05

0.0001

0.001

0.01

0.1

1

0 2 4 6 8 10

KE

y

Time

2*Γ = 4.84


1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 2 4 6 8 10

KE

y

Time

2*Γ = 5.9


1e-06

1e-05

0.0001

0.001

0.01

0.1

0 2 4 6 8 10

KE

y

Time

2*Γ = 6.3


1e-06

1e-05

0.0001

0.001

0.01

0.1

0 2 4 6 8 10

KE

y

Time

2*Γ = 6.04


1e-07

1e-06

1e-05

0.0001

0.001

0.01

0 2 4 6 8 10

KE

y

Time

2*Γ = 5.24


1e-07

1e-06

1e-05

0.0001

0 2 4 6 8 10

KE

y

Time

2*Γ = 3.87


1e-07

1e-06

1e-05

0 2 4 6 8 10

KE

y

Time

2*Γ = 2.5

"k_8/Energy.dat" u 1:3"k_8/Energy.dat" u ($1)-0.2:(exp(2.5*($1)))*2e-07

• Lx = Ly = 2π; ω0 = 25; d = 3π128 ; ν = 0.0015



33/21



Growth Rate Comparison of KH with AGSpect; k = 3

1e-06

1e-05

0.0001

0.001

0 20 40 60 80 100

KE

y

Time

C-FD2D

AGSpect



34/21



Growth Rate with Mach Number

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

1 2 3 4 5 6 7 8

Gro

wth

Rate

(2

γ)

Mode Number (m)

Analytical and Numerical Growth Rate with Compressibility

"Drazin.dat" u 1:2"slope.dat" u 1:2"slope.dat" u 1:3"slope.dat" u 1:4"slope.dat" u 1:5"slope.dat" u 1:6"slope.dat" u 1:7"slope.dat" u 1:8

Figure: Analytical & M = 0, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5



35/21



Benchmarking of MHD2D for compressible flows

Growth Rate at Large Mach Number: M = 0.5

-25

-20

-15

-10

-5

0

0 1 2 3 4 5 6 7 8 9 10

ln E

y

t

fig2 Keppens

"Energy_2p5.dat" u 1:(log($4))"Energy_2p5.dat" u ($1):(3.4752*($1))-24.5

Figure: Benchmarked with R Keppens et. al., JPP, 61 (1999)



36/21


Benchmark of MHD2D: MagnetoHydroDynamic (KH)

Benchmarking of compressible MHD flows

Growth Rate at Large Mach Number: M = 0.5

0

0.05

0.1

0.15

0.2

0.1 0.2 0.3 0.4 0.5 0.6

Gro

wth

Ra

te (

γ a

/ V

0)

kx a

Hydrodynamic

Magnetohydrodynamic

Figure: Benchmarked with R Keppens et. al., JPP, 61 (1999)



37/21


Benchmark of MHD2D: MagnetoHydroDynamic (KH)

Benchmarking of MHD2D for compressible flows

For compressible Vortex Merger Problem with AGSpect

0.089

0.09

0.091

0.092

0.093

0.094

0.095

0 20 40 60 80 100

Kin

etic

E

ne

rg

y

t

M = 0.05M = 0.5

0

2e-05

4e-05

6e-05

8e-05

0.0001

0.00012

0.00014

0.00016

10 20 30 40 50 60 70 80 90 100

Po

we

r

Frequency (ω)

AGSpectC-FD-2D

0

2

4

6

8

10

12

0 2 4 6 8 10 12

t = 0"M_0p5/fort.100" u 1:2:3

0

2

4

6

8

10



38/21


Numerical checks of MHD2D for KH flow in MHD system

~∇ · ~B = 0 = ~∇ · ~u

-6e-32

-4e-32

-2e-32

0

2e-32

4e-32

6e-32

0 2 4 6 8 10-2e-11

-1.5e-11

-1e-11

-5e-12

0

5e-12

1e-11

1.5e-11

2e-11

Div

B

Div

V

time

Ideally Divergence B = 0

Divergence B

Divergence V



39/21


GPU Performance of G-MHD3D (With Nagavijayalakshmi Vydyanathan, NVIDIA, India)



40/21


GPU Performance of G-MHD3D with Passive Tracers (With Vinod Saini, IPR & Naga Vydyanathan, NVIDIA, India)



some novel features of three dimensional ...rupak mukherjee rajaraman ganesh abhijit sen institute...

Documents