Seminar Slide Deck (PDF-7.2MB)

Compressibility Effects on Dielectric Barrier Discharge Actua9on and Boundary Layer Recep9vity

Marie Denison, Luca Massa University of Texas at Arlington, TX

Nasa Applied Modeling & Simula3on (AMS) Seminar Series NASA Ames Research Center, Moffe> Field, CA April 29, 2014

Outline

•  Introduc9on •  Linear Stability Analysis Framework •  AMR Coupled Plasma-‐Navier and Stokes Solver •  Discharge Features •  Recep9vity Analysis and Design Implica9ons •  Summary

2

Outline


3

Mo9va9on

4

Supersonic flow boundary layer instabili9es •  Cross-‐flow dominated laminar-‐turbulent transi9on along supersonic cones and swept wings •  Turbulence and detachment in scramjet inlets, upstream shock propaga9on, unstart

Synthe9c plasma roughness •  Control of distributed Dielectric Barrier Discharge electrodes adapted to flight condi9ons •  For example subcri9cal forcing of sta9onary waves or boundary layer ernergiza9on / thinning

Figures of Merit of DBD actua9on

5

FOM DBD Solid Roughness

Control yes: height, thrust, ac9va9on paWern, switch-‐off

no

BL penetra9on volume forcing, horizontal/ver9cal jets

low

Wall thermal conduc9vity

low (<1.5W/Km)

depends

Power electric supply, temporary drag

Drag

Weight larger (mat. density, electrodes, wiring, supply, control)

lower

Reliability dielectric breakdown, temperature limit, electrode wear-‐off

surface wear-‐off

Manufacturing effort and cost

larger lower

TRL low high

kV or float

Ground

Challenges

6

Y.C. Schuele, Ph.D., Univ. of Notre Dame, 2011

Physics: •  DBD atmospheric gas chemistry •  Hea9ng, real gas effects •  Surface reac9ons •  Plasma-‐Flow energy coupling (Te, Tg, ηth, Da) CFD and Recep9vity Analysis: •  "Pulsing roughness" •  3D Micro-‐filamenta9on •  Sta9s9cal characteris9cs, rela9on to surface proper9es •  Length (~μm cathode sheath) and 9me scales (ps...s) è model reduc9on •  HPC, automated refinement around discharges •  Experimental valida9on of gas models (flow on/off)

This Work

System •  2D Flat plate •  3-‐species Helium gas •  Eigenmode growth (Tollmien-‐Schlich9ng) Research •  Study of compressibility effects

–  on DBD discharge features –  on linear recep9vity to DBD perturba9on

•  New coupled AMR Plasma Navier-‐Stokes solver (1-‐3D) •  Adjoint based formula9on of the compressible boundary layer

recep9vity problem 7

This Work

8

2D Flat Plate Helium

M=0.5-2.0

Naver-Stokes BL Linear Stability

Navier-Stokes/Plasma AMR Solver

Direct & Adjoint operators

Modal Solutions (ω, M, Re)

Gas

Governing Equations

Geometry

Receptivity Analysis

Flow and Force effects

Discharge Features

Solver Structure

Force(x,y)

Integrated Source Term

Peak Velocity

Receptivity Coefficients

Design Aspects

Receptivity Coefficient

Mesh Convergence

Outline


9

Linear Stability Framework

•  DBD induced velocity ~m/s << u∞

•  Quasi-‐parallel approxima9on •  {M, Re} range to support TS waves •  Non-‐dimensionaliza9on, a.o.

•  Horizontal velocity perturba9on from f(x,y)exp(-‐iωt) source term and discrete modes

where Ak is the recep9vity coefficient

10

0 1 2 3 4 5 610−1

100

101

102

103

(Rex)−1/2× 1000

f (kH

z)

M=0.5M=0.8M=1.2M=2.0

u = Akφk (y)e− i(ω kt−α kx )

k∑

L* = xRex

, Rex =ρu∞

* xµ∞* ,ω =ω * L*

u∞*

Ak =dx f (x, y)φ̂k (y)e

− iα kx dy0

∞

∫a

b

∫[φk ,φ̂k ] xa xb

y

Source f(x,y)

Adjoint Eigenproblem

•  in Ak is the adjoint mode to , obtained from the solu9on of the adjoint system itself derived from the Euler-‐Lagrange's iden9ty

•  Interpreta9on:

–  Assume point unit force f(x,y)=δ(x0,y0), –  Normalize by max|ϕk (y)| and by

–  The adjoint mode propaga9on velocity is opposite to the regular mode è for Re(α)>0, mode amplifica9on is upstream towards the source

•  The method allows tes3ng mul3ple sources using the solu3ons of the homogeneous regular and adjoint eigenvalue problems

11

φ. Λ ∂φ̂∂t

+ L̂(φ̂)⎛⎝⎜

⎞⎠⎟+ φ̂. f = ∂Γ(φ,φ̂)

∂t+∇.J(φ,φ̂)

φ̂k φk

Ak = φ̂k (y0 )

φ̂k [φk ,φ̂k ]= 1φk

Regular and Adjoint Mode Solu9ons

12

M∞=0.5 M∞=2.0

•  Chebyshev τ-‐QZ method used to solve the eigenproblems •  At high M∞, the adjoint mode (unit-‐point-‐force sensi9vity) decreases and the

depth of its maximum increases

Comparison to incompressible case

13

•  D.C. Hill's results (J. Fluid Mech. '95) data well reproduced @ M∞=0.1 (lines) •  At M∞=0.8 (symbols), the mode depth increases and its amplitude decreases F= ω/√(Rex)

Adjoint modes

•  Up to 3x depth increase and 1/9x amplitude decrease over M∞ range •  Recep9vity benefit in matching the source and adjoint profiles

14

0 0.05 0.1

1

2

3

4

5

6

y @ m

ax. a

mplitu

de

ω

M=0.5

Rex=2x105

Rex=4x105

Rex=8x105

0 0.05 0.10

2

4

6

8

10

12

max.

ampli

tude

ω

M=0.5

Rex=2x105

Rex=4x105

Rex=8x105

0 0.05 0.1

1

2

3

4

5

6

ω

M=0.8

Rex=2x105

Rex=4x105

Rex=8x105

0 0.05 0.10

2

4

6

8

10

12

ω

M=0.8

Rex=2x105

Rex=4x105

Rex=8x105

0 0.05 0.1

1

2

3

4

5

6

ω

M=1.2

Rex=2x105

Rex=4x105

Rex=8x105

0 0.05 0.10

2

4

6

8

10

12

ω

M=1.2

Rex=2x105

Rex=4x105

Rex=8x105

0 0.05 0.1

1

2

3

4

5

6

ω

M=2

Rex=2x105

Rex=4x105

Rex=8x105

0 0.05 0.10

2

4

6

8

10

12

ω

M=2

Rex=2x105

Rex=4x105

Rex=8x105

Student Version of MATLAB

Outline


15

System and Regimes

•  2D Flat plate –  1mm dielectric thickness, 3.5 dielectric constant –  1.2cm electrode width –  2.45cm distance between downstream plate edge and electrode edge...

•  Flow –  M∞ = 0.5 to 2.0 –  Re = 2x105 to 8x105 –  Self-‐similar profile at inflow boundary –  ... Momentum thickness for effec9ve xd

•  Gas –  Simplified helium chemistry (He, He+, e-‐) –  Constant electron temperature (1eV) –  impact ioniza9on and recombina9on rates per S. Roy, Phys. of Plasmas, Vol. 13, '06 –  Reduced electric field dependent mobili9es

€

xd = Re µ∞

ρ∞u∞

1.2cm

1.2cm

1mm x

y

16

AMR Solver

•  Governing equa9ons

•  Wall boundary condi9ons –  2,828V at the cathode, f = 5kHz –  adiaba9c surface –  secondary emission at the cathode (coeff.=0.26) –  thermal flux and relevant charged species dri� towards the wall –  gas/dielectric interface charge from the species flux towards the wall Dielectric displacement by stencil manipula9on at the 1D embedded boundary

17

continuity: ∂nj∂t

+∇.Fj = Gj ,kk∑

drift-diffusion: Fj = nju j = sign(Z j )njµ jE− Dj∇nj + njVj

Poisson: ε ∇.E = Z jqjnjj∑ E E

n

i

e

n

i

e

dσ s

dt= qZ jFj

j∑ .n, εdEd − εgEg( ).n =σ s

AMR Solver

•  Univ. of Berkeley's Chombo Adap9ve Mesh Refinement C++ numerical framework for the solu9on of PDEs –  block structured –  embedded boundary –  hierarchy of rectangular la�ces

communica9ng through ghost cells –  graph represen9ng the irregular cells –  physics-‐based automated tagging and

refinement •  Two independent meshes for flow and

plasma, synchronized at coarse 9me steps •  Implemented on Texas Advanced Compu9ng

Center (TACC), Stampede HPC system 18

P. Collela et al, Chombo SoYware Package for AMR Applica3ons Design Document, March 2012

19

Plasma SolverLevel operations for

advection, diffusion and reactions

Navier-Stokes SolverLevel operations for continuity,

momentum and energy

Initialization- Similarity boundary layer u, v, ρ, !- Ne, N(background)- Tg, (Te, ee)

Gamma GasPolytropic approximation

Godunov method with van Leer splitting

Viscous fluxesSutherland viscosity relationCell centered finite difference

AdvectionGodunov with

van Leer splitting

Diffusive termsImplicit backward Euler

Electric field

Body forcesGas source energy

Time advance

Velocity, P, Tg

Plasma PhysicsGas model

Reaction rates Chemical source terms

(Electron source energy)Transport coefficients Wall boundary fluxes

Species concentrations(Electron Energy ee, Te)

Wall charge

Reactive terms5th order stiff ODE

solver

ODE solver

AMR solver

Adap9ve Mesh Refinement

•  M∞=0.8, Re=0.8x106 •  7 refinement levels for the species

–  thresholds: E>2.5x106 V/m, |F|>250N/m3, ne=1x10-‐9mol/m3

•  4 refinement levels for vor9city, threshold |ω|/u∞>200m-‐1 •  CFL in the last periods = CFLa*(1+M∞)~0.28

20

Convergence

•  Periodic solu9on a�er 2-‐3 periods in studied M∞ range •  Solu9on converged within 2-‐5% with 7 species levels •  Max. nr of grid points at ~ 0.7 T (@Vmax), following the induced velocity

3.3 3.4 3.5 3.6 3.7 3.8 3.9 40.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

t/Tperiod

Max

stre

amw

ise

velo

city

Δ xmin = 15.625 µΔ xmin = 7.8125 µΔ xmin = 3.9063 µ

3.2 3.4 3.6 3.8 4 4.20

1

2

3

4

5

6

7 x 105

t/Tperiod

Num

ber o

f grid

poi

nts

Δ xmin = 7.8125 µΔ xmin = 3.9063 µ

21

Outline


22

Integrated source term

23

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6x 10−3

t/Tperiod

Inte

grat

ed s

ourc

e [N

/m]

M = 1.2

Rex = 2 105

Rex = 4 105

Rex = 8 105

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

x 10−3

t/Tperiod

Inte

grat

ed s

ourc

e [N

/m]

Rex = 4 × 105

M∞=0.5M∞=0.8M∞=1.2M∞=2

•  Monotonic integrated source term reduc9on with decreasing Reynolds nr and increasing M∞

Sint fx2 + fx

2 dA N/m[ ]A∫

24

•  Non-‐monotonic Reynolds effect with increasing M∞

Peak velocity

0 0.2 0.4 0.6 0.8 10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

t/Tperiod

Max

stre

amw

ise

velo

city

, [m

/s]

M∞ = 0.5

Rex = 2 105

Rex = 4 105

Rex = 8 105

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t/Tperiod

Max

stre

amw

ise

velo

city

, [m

/s]

M∞ = 1.2

Rex = 2 105

Rex = 4 105

Rex = 8 105

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t/Tperiod

Max

stre

amw

ise

velo

city

, [m

/s]

M∞ = 0.8

Rex = 2 105

Rex = 4 105

Rex = 8 105

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t/Tperiod

Max

stre

amw

ise

velo

city

, [m

/s]

Rex = 4 × 105

M∞=0.5M∞=0.8M∞=1.2M∞=2

Velocity Profile

25

M∞=0.5 M∞=0.8

M∞=1.2 M∞=2.0

Force profile

•  5 countours at Sx = {1 × 103,5 × 103,1 × 104,5 × 104,1 × 105} N/m3 •  Profile flaWening/elonga9on with increasing M∞

26

Outline


27

Force FFT Decomposi9on

•  Model decomposi9on with FFTW •  200 samples per actua9on period •  steep amplitude decrease with f

28

0 10 20 30 40 50

0.5

1

1.5

2

2.5

3

x 10−3

f, [kHz]

S int [N

/m]

M∞ = 0.5M∞ = 0.8M∞ = 1.2M∞ = 2

M∞=0.5, Rex=400k, ω*=2fDBD

M∞=2.0, Rex=400k, ω*=9fDBD

y*/L*

y*/L*

Abs f x, y( )φn '(y)e− iαx( )

Sint fx2 + fy

2 dA N/m[ ]A∫

Lengthscales

•  Note decrease of L* and xd with M∞ and decreasing Rex

29

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

M

Cat

hode

edg

e po

sitio

n [m

]

Rex=2x105

Rex=4x105

Rex=8x105


0 0.5 1 1.5 2 2.50

1

2

3

4

5

x 10−4

M

L* [m]

Rex=2x105

Rex=4x105

Rex=8x105


FFT Frequency spectrum

•  More modes par9cipate with increasing M∞ (see symbols)

•  Including larger mul9ples of ωDBD (i.e., with smaller integrated source)

30

0 0.02 0.04 0.06 0.080

0.002

0.004

0.006

0.008

0.01

ω

−im

ag(α)

M=0.5

Rex=2x105

Rex=4x105

Rex=8x105

0 0.02 0.04 0.06 0.080

0.002

0.004

0.006

0.008

0.01

ω

−im

ag(α)

M=0.8

Rex=2x105

Rex=4x105

Rex=8x105

0 0.02 0.04 0.06 0.080

0.002

0.004

0.006

0.008

0.01

ω

−im

ag(α)

M=1.2

Rex=2x105

Rex=4x105

Rex=8x105

0 0.02 0.04 0.06 0.080

0.002

0.004

0.006

0.008

0.01

ω

−im

ag(α)

M=2

Rex=2x105

Rex=4x105

Rex=8x105


ω

ω =ω * L*

u∞*

Modal Kine9c Energy

•  Kine9c energy independent of mode k normaliza9on

•  Symbols show the most amplified mode max(Im(-‐α))

•  Non-‐monotoneous response to

Rex. Decreased effect at M∞=1.6

31

KEk = Ak2ρ

0

∞

∫ φ̂k†(y)φk (y)dy

0.02 0.04 0.06

10−1

100

101

102

ω

√KE

1x1

05

M=0.6

Rex=2x105

Rex=4x105

Rex=8x105

0.02 0.04 0.06

10−1

100

101

102

ω

√KE

1x1

05

M=0.8

Rex=2x105

Rex=4x105

Rex=8x105

0.02 0.04 0.06

10−1

100

101

102

ω

√KE

1x1

05

M=1.2

Rex=2x105

Rex=4x105

Rex=8x105

0.02 0.04 0.06

10−1

100

101

102

ω

√KE

1x1

05

M=1.6

Rex=2x105

Rex=4x105

Rex=8x105


Effect of Force

•  Using the force FFT components at M∞=0.8 for M∞=1.2, 1.6 •  Rela9vely insensi9ve recep9vity response to M-‐varia9on, as expected

from the slight force profile change observed on slide #26

32

0.01 0.02 0.03 0.04 0.05 0.0610−2

10−1

100

101

ω

√KE

1x1

05

Rex=4x105

M=0.8M=1.2, Force @ M=1.2M=1.2, Force @ M=0.8M=1.6 @ Force M=1.6M=1.6 @ Force M=0.8


Most Amplified Mode at fDBD=5kHz

•  Bands @ ωmax,TS/ωDBD=const •  Within each band, √KE increases along δM∞/δRex<0

due to an increasing overlap of the adjoint mode with the force

33

0.4 0.6 0.8 1 1.2 1.4 1.6 1.810−2

10−1

100

101

M

√KE

1x1

05 @ m

ax(−α)

Rex=2x105

Rex=4x105

Rex=8x105


dark blue 1/2 log(KE) = −14.7 dark red 1/2 log(KE) = −12.7

Linear interpola3on

Force Mode Shape Effect

•  Using 1st harmonic mode (symbols) instead of actual harmonic @ max(Im(-‐α)) •  Suggests benefit in selec9ng DBD frequency at the most amplified mode •  Possible trade-‐off as DBD thrust diminishes with frequency

34

0.4 0.6 0.8 1 1.2 1.4 1.6 1.810−2

10−1

100

101

M

√KE

1x1

05

Rex=4x105, ω @ max(−α)

Force(max(−α))Force(1st harmonic)


Outline


35

Summary

36

•  An analysis of compressible effects on flow recep9vity to DBD actua9on in a 2D set up with He gas was presented.

•  The dependence of the force, peak velocity and integrated source term on flow condi9ons was inves9gated.

•  The main findings in terms of recep9vity can be summarized as follows •  Recep9vity decreases between M∞=0.5-‐2, with satura9ng Rex effect •  For an an increase in M, an actuator shi� upstream is beneficial to increase

the overlap with the highly recep9ve region •  For the gas system under considera9on, the flow dependence of the force

has liWle effect on recep9vity. •  DBD frequency matching to the most amplified mode may increase

recep9vity in a frequency range where the force does not degrade much •  A new coupled plasma – Navier and Stokes solver was developed. •  The AMR feature allows for dynamic tracking of the discharge into the volume of

the flow, while the embedded boundary capability allows simula9ng complex 3D geometries.

Future Research

•  Reduced air chemistry, with most relevant species/reac9ons may lead to different conclusion on flow-‐plasma interac9ons

•  Accelerated numerical models addressing the s9ffness of fast chemical reac9ons

•  3D cross-‐flow linear, non-‐linear recep9vity and transi9on studies with plasma roughness elements.

37

Acknowledgements

•  Donald A. Durston and Ce9n C. Kiris •  Esteban Cisneros, Grad. Student in Aerospace Engineering at the

University of Texas at Arlington, TX •  TACC (Texas Advanced Compu9ng Center) at UT Aus9n This presenta9on includes material from the following publica9ons: AIAA 2014-‐0485, AIAA 2012-‐2737

38

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