slides kth stability and transition

7/24/2019 Slides KTH stability and transition

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Non modal stability

!"#$ &'$()* + ,-*-' .#/01)

!1((2 3!45 6-(*'-7 89: ;-#/$(1#-'1$? 6@??-A- !@()@(7 B8


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4"*?1(-

.*$C1?1*D @E F"1)


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$( -H$0>?- @E 1(


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P"0-'1#$?


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61''"< #?@")< )-J-?@>1(A 1( $ L-* $#- ./"V?- RPK.KT


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K( -H$0>?- @E 8-?J1(M:-?0@?*W


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9'$(1>- F@N -H>-'10-(* RXYYZT


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9'$(1>- F@N -H>-'10-(* RXYYZT


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9'$(


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6?$


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&D>$


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4-( F@N


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9N@ #@(#->*< @E


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.*$C1?1*D $($?D


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:D)'@)D($01#


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.*$C1?1*D

.*$C?- P-"*'$?

B(


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4J-'J1-N

.*$C1?1*D $($?D


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.*$C1?1*D $($?D'@C?-0

U = (U , V , W ) , T =T(xi)

u

t= F(u;U)

u

t = A(U)u

u(xi, t) = u eit

iu= A(U)u

u(xi, t), T(xi, t)


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!1(-$' 0 i = 0 i < 0

B(


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9/-'0$? 1(


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9-'01


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9-'01$(


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x or y

z

T0

T0 T

d

2(0&"%3456"-(.7 %-/'(*%&%'0


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2(0&"%3456"-(.7 %-/'(*%&%'0

Linear stability theory: above a critical Rayleigh number of

1708 the conductive state becomes unstable to infinitesimal

perturbations

Energy stability theory: below a critical Rayleigh number of

1708 finite-amplitude perturbations superimposed on the

conductive state decay monotonically in energy

Rayleighnumber: ratio between buoyancy forces (temperaturegradient) and viscous forces the governing parameter


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2(0&"%3456"-(.7 %-/'(*%&%'0

Linear stability theory: above a critical Rayleigh number of

1708 the conductive state becomes unstable to infinitesimal

perturbations

Energy stability theory: below a critical Rayleigh number of

1708 finite-amplitude perturbations superimposed on the

conductive state decay monotonically in energy

Rayleighnumber: ratio between buoyancy forces (temperaturegradient) and viscous forces the governing parameter

89:".%1"-'/ /4+$ '4" +-/"' +;


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C&(-" C+%/",%&&" D+$

Reynoldsnumber: ratio between inertial forces and viscous forces thegoverning parameter


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C&(-" C+%/",%&&" D+$

Linear stability theory: above a critical Reynolds number of

5772 the parabolic profile becomes unstable to infinitesimal

perturbations

Energy stability theory: below a critical Reynolds number of

49.6 finite-amplitude perturbations superimposed on the

parabolic profile decay monotonically in energy



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C&(-" C+%/",%&&" D+$

Linear stability theory: above a critical Reynolds number of

5772 the parabolic profile becomes unstable to infinitesimal

perturbations

Energy stability theory: below a critical Reynolds number of

49.6 finite-amplitude perturbations superimposed on the

parabolic profile decay monotonically in energy


89:".%1"-'/ /4+$ ',.*,&"-' :('


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9N@ @>>@


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Energy equation


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Linear growth mechanisms


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."C#'1G#$? *'$(


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."C#'1G#$? *'$(


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."C#'1G#$? *'$(


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P@(M0@)$? $>>'@$#/


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!1(-$' '@C?-0

.*$'* E'@0 ?1(-$'1@(-(G$?

U

d

dtq = Lq

q= exp(tL)q0; q(t= 0) = q0


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P@'0 @E 0$*'1H -H>@(-(G$?

=(>"* @"*>"* $($?D


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P@'0 @E 0$*'1H -H>@(-(G$?

=(>"* @"*>"* $($?D


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;$*'1H (@'0

m"#?1)-$(


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m1A-(J$?"-< J< ,'@>$A$*@'P@'0

;$*'1H -H>@(-(G$? )1l#"?* *@ #@0>"*-

.D


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m1A-(J$?"-< J< ,'@>$A$*@'P@'0

B>>-' $() ?@N-' C@"()< @E

Lower bound

G(t)

e2tmax || exp(tL)||2

|| exp(tL)||2 =|| exp(tSS1)||2

||S||2||S1||2e2tmaxUpper bound

9/- -(-'AD #$((@* )-#$D $* $ E$


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&@"()< @E */- 0$*'1H -H>@(-(G$?

e2tmax || exp(tL)||2 ||S||2||S1||2e2tmax

Two distinct cases:

(S) =||S||2||S1||2Condition number:

(S

) = 1

,::". (-7 &+$". *+,-7


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&@"()< @E */- 0$*'1H -H>@(-(G$?

e2tmax || exp(tL)||2 ||S||2||S1||2e2tmax

Two distinct cases:

(S) =||S||2||S1||2Condition number:

,::". (-7 &+$". *+,-7


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P@(M(@'0$?1*D

(S) =||S||2||S1||2 = 1

(S) =||S||2

||S1

||2

1

H+.1(& /'(*%&%'0 :.+*&"1E

@'*/@A@($? -1A-(J-#*@'$


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6?$


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&D>$


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4>G0$? 1(1G$? #@()1G@(

Initial condition that results in the maximumenergyamplification at a given time

||q(t)||2 =||q0||2 = 1

q(t) = exp(tL)q0

exp(tL) q0 = || exp(tL)|| q(t)

>'@>$A$*@' 1(>"* $0>?1]#$G@( @"*>"*


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4>G0$? 1(1G$? #@()1G@(

exp(t

L) q0 = || exp(t

L)|| q(t

)

>'@>$A$*@' 1(>"* $0>?1]#$G@( @"*>"*

Singular value decomposition of a matrix

V = UA = UVH

Au1v1

1 = ||A||2


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4>G0$? 1(1G$? #@()1G@(

exp(t

L) q0 = || exp(t

L)|| q(t

)

>'@>$A$*@' 1(>"* $0>?1]#$G@( @"*>"*


u1v1

svd(exp(tL)) = UVH

exp(tL)

G(t) =|| exp(tL)||


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4>G0$? 1(1G$? #@()1G@(

exp(t

L) q0 = || exp(t

L)|| q(t

)

>'@>$A$*@' 1(>"* $0>?1]#$G@( @"*>"*


u1v1

svd(exp(tL)) = UVH

exp(tL)

G(t) =|| exp(tL)||

Optimal initial conditionleft principal

singular vector

Optimal final conditionright principalsingular vector


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4>G0$? )1


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Lift-upmechanism in shear layers(Ellingsen & Palm 1975, Landahl 1980)

Particle displaced in thewall-normal directionretain their horizontal

momentum

Streamwise vorticesinduce streamwise

streaks

In boundary layers: wall-normal shear is O(Re)

streak growth O(Re)

Streamwisevortices

Streamwisestreaks

P@(M;@)$? ^'@N*/


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4"*?1(-

.*$C1?1*D @E F"1)


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I-#->GJ1*D

=(*-'-


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4>G0$? '-@(


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&@"()< @E '-


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&@"()< @E '-


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&@"()< @E '-


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4>G0$? E@'#1(A

.1(A"?$' J$?"- )-#@0>@"*=(>"*

||qp|| =|| f|| = 1


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I-


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I-


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6@0>@(-(*MN1


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6@0>@(-(*MN1"*M@"*>"* $($?D


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K)L@1(*


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K)L@1(*


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mH>$(


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BGJ1*D

ut

+L(U,Re)u+p= 0

u = 0

Linearized Navier-Stokes(u, p)

(f+

,m

+

)Differentiable fields

0 = u

t

+L(U,Re)u+pf+ + ( u)m+Sum and multiply

Integrate by parts overtime and space


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BGJ1*D

Integrate by parts over time and space

L(U,Re)u= U u+ u U 1

Re

2u

L+

(U,Re)f+

=U

f+

U

f+

+

1

Re2

f+

J=U(uf+) + 1

Re(f+ u uf+) +m+u+pf+

where

Z0

ZD

u

t +L(U,Re)u+p

f+ + ( u)m+

=

Z t

0

ZD

u

f+

t +L+(U,Re)f+ +m+

+p( f+)

+

Z t

0

uf+

t +

ZD

J


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BGJ1*D

Integrate by parts over time and spaceZ0

ZD

u

t +L(U,Re)u+p

f+ + ( u)m+

=

Z t

0

ZD

u

f+

t +L+(U,Re)f+ +m+

+p( f+)

+

Z t

0

uf+

t +

ZD

J

Definition of adjoint problem

Assume volume forcing, mass source and integrate in timeu

t+L(U,Re)u+p= f u = Q

u(t) f+(t) u(0) f+(0) =

Z t

0

ZD

f f+ + Qm+

+

ZD

J n


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BGJ1*D

u(t) f+(t) u(0) f

+(0) =Z t0

ZD

f f

+ + Qm+

+ZD

J n

Assume initial condition for adjoint system f+

(t) =u(t)

u(t) u(t) =u(0) f+(0) +

Z t

0

ZD

f f+ + Qm+

+

ZD

J n

E ) ] ?) E


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u(t) u(t) =u(0) f+(0) +

Z t

0

ZD

f f+ + Qm+

+

ZD

J n

BGJ1*D

u(t) f+(t) u(0) f

+(0) =Z t0

ZD

f f

+ + Qm+

+ZD

J n


(t) =u(t)

djoint velocity gives sensitivity to initial condition and forcing

u2(t)

u(0)

=f+(0) u2(t)

f

=f+

E ) ] ?) E


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u(t) u(t) =u(0) f+(0) +

Z t

0

ZD

f f+ + Qm+

+

ZD

J n

BGJ1*D

u(t) f+(t) u(0) f

+(0) =Z t0

ZD

f f

+ + Qm+

+ZD

J n


(t) =u(t)

djoint pressure gives sensitivity to mass source

u2(t)

Q

=m+

E )L 1 ] ?) E G 1


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u(t) u(t) =u(0) f+(0) +

Z t

0

ZD

f f+ + Qm+

+

ZD

J n

BGJ1*D

u(t) f+(t)

u(0) f

+(0) =Z t0

ZD

f f

+ + Qm+

+ZD

J n


(t) =u(t)

Gradient of adjoint field gives sensitivity to boundary

conditions

u2(t)

uwall=

1

Ref+ +m+ n

4 ?1


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4"*?1(-

.*$C1?1*D @E F"1)


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.*'"#*"'$? $'$0-*-'QI-D(@?)< ("0C-'7 C$


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.*'"#*"'$? $'$0-*-'QI-D(@?)< ("0C-'7 C$


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.*'"#*"'$? $'$0-*-'QI-D(@?)< ("0C-'7 C$


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.*'"#*"'$?


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.*'"#*"'$?


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.-(


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.-(


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3?@N #/$'* E@'


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3?@N $'@"() $ #D?1()-'

XO C$


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3?@N $'@"() $ #D?1()-'

XO C$


98/172

3?@N $'@"() $ #D?1()-'

XO C$


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3?@N $'@"() $ #D?1()-'

XO C$


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3?@N $'@"() $ #D?1()-'

XO C$


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3?@N $'@"() $ #D?1()-'

XO C$


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4-( F@N


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D D

Oscillators !Modal analysis

M !$'A-G0$? E@'#1(A R>


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^?@C$??D "(>'@$#/ 1)-(G]-< $-#1]# $G$? >@


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!$A'$(A- 1)-(G*D $()

$)L@1(* -_"$G@( AD >

./-))1(A 0@)- =Q N$J-0$U-' J


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6@(-'$*@' -H>'-


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6@(-'$*@' -H>'-


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.*'"#*"'$? J$'1$G@(< @E */- C$


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5$U- #@(*'@? CD 0-$(< @E


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;@)- =

.-(


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;@)- =

.-(


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;@)- =

9@*$? .-(


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;@)- ==

.-(


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;@)- ==

.-(


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;@)- ==

9@*$? .-(


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.10"?$G@(< @E $ @


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P@1?1]-''-G0$? '-@(?1]-'

&'$()* -* $?O7 `@"'($? @E 3?"1) ;-#/O7 [\XX

^'$)1-(* @E */- '-


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4>G0$? E@'#1(A $() '-@( @E @>G0$? E@'#1(A $() '-@(


123/172

F247%" 1#)9,*.G ?,H@I2 -J-9(

#.*'-$0N1?$*-


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F247%" 1#)9,*.G KA@/%:-+

#6@(J-#GJ- (@(M(@'0$?1*D7 [h E@'#1(A $* E'-_"-(#D 3cX\\O

#3@'#1(A $#GJ- C-*N--( C'$(#/ = $() ==

Brandt et al, JFM, 2011

.-(


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9.MN$J-

^-(-'$?1W- *@ G0-M>-'1@)1# $() G0-M)->-()-(* F@N< I-?$*- (@(M0@)$? $($?DG01W$G@( >'@C?-0"*M@"*>"* $($?D


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> > D

.--U @>G0$? -(-'AD $0>?1]#$G@(

G(t) = maxq0

hq, qi

hq0, q0i

=(>"*M@"*>"* $($?D


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> > D

.--U @>G0$? -(-'AD $0>?1]#$G@(

G(t) = maxq0

hq, qi

hq0, q0i

= maxq0

hA(t)q0, A(t)q0ihq0, q0i

=(>"*M@"*>"* $($?D


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> > D

.--U @>G0$? -(-'AD $0>?1]#$G@(

G(t) = maxq0

hq, qi

hq0, q0i

= maxq0

hA(t)q0, A(t)q0ihq0, q0i

= maxq0

hAH(t)A(t)q0, q0i

hq0, q0i

=(>"*M@"*>"* $($?D


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E@' G0-M)->-()-(* "* $($?D


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E@' G0-M)->-()-(* '@C?-0


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E@' G0-M)->-()-(* '@C?-0


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$)L@1(*

>'@C?-0

(n)

)1'-#*>'@C?-0

'@C?-0


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g$'1$G@($? E@'0"?$G@( @E */- @>G0$? A'@N*/ >'@C?-0more general

J =

||q||2

||q0||2 maxWe wish to maximize

with the constraintd

dt

q=L(t)q

Listen Carlo Cossu on Friday!

^?@C$? 0@)-


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;@'@C?-0< N1*/ 1(/@0@A-(-@"< )1'-#G@(?-H A-@0-*'D

6$((@* "G0$?-() @( 0@'- */$(

@(- )1'-#G@( u+(x, y)

6@0>"*$G@($? 1


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One vs two inhomogeneous directions:state vector, matrix, operation count

L CN

2N

2

O(N4)L CNN

O(N2)

O(N3) O(N6)

Matrix size

Operation count

State vectorq=

q1

q2

qN

q=

q1,1

q1,2

qN,N

6@0>"*$G@($? 1


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One vs two inhomogeneous directions:state vector, matrix, operation count

" Direct eigenvalue algorithms become too expensive

"

Iterative algorithms,Arnoldi technique

L CN

2N

2

O(N4) O(N6)

storage

CPU time

K'(@?)1 $?A@'1*/0


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K#G@( @E */- ?1(-$' @>-'$*@' N1*/1( $( @'*/@(@'0$? C$


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K#G@( @E */- ?1(-$' @>-'$*@' N1*/1( $( @'*/@(@'0$? C$'-


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Only multiplication by L are necessary

System eigenvalues approximated

by eigenvalues of H

Eig(L) Eig(H)

mH$0>?- 1( [hQ F@N 1( sML"(#G@(


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Base flow

Lashgari et al, in preparation

mH$0>?- 1( [hQ F@N 1( sML"(#G@(


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Spectrum


" Steady two-dimensional bifurcation

mH$0>?- 1( [hQ F@N 1( sML"(#G@(


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Eigenfunction


"

Steady two-dimensional bifurcation

mH$0>?- 1( [hQ F@N 1( sML"(#G@(


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New asymmetric state


"

Steady two-dimensional bifurcation

sML"(#G@(Q mH$0>?- 1( [7iha


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Asymmetric state unstable to 3D periodic disturbances

Snapshot method based on linear DNS

mH$0>?- 1( ZhQ `-* 1( #'@


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Bagheri et al, JFM, 2011, Schlatter et al.

Use DNS and compute spectrum of matrix exponentialq= exp(tL)q0

."00$'D


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.*$C1?1*D @E F"1) '@C?-0-()-(* >'@C?-0Q -1A-(J$?"- >'@C?-0 1( G0-O mHb

6?$


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,$'$??-?


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4''M.@00-'E-?) $() ._"1'- -_"$G@(


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P@(0@)$?


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=(>"*M@"*>"* $>>'@$#/

P@((@'0$? @>-'$*@'

slides kth stability and transition

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