Linear hydrodynamic stability analysisand print-quality in high-speed ink-jets

G. D. McBain & S. G. Mallinson

Simulation & ModellingMemjet Australia Pty Ltd

Macquarie Park

FluDThe University of Sydney

Mon. 20 Mar. 2017

Part I.One-dimensional linear hydrodynamic stability analysis

G. D. McBain

What is linear stability analysis?

Consider a dynamical system: a set of unknowns x evolve in time taccording to an autonomous differential–algebraic equation:

f (x , v) = 0 , for v ≡ dx

dt

Say x = X is a steady solution:

f (X , 0) = 0 .

Consider a perturbation x ∼ X + εξ:

f

(X + εξ, ε

dξ

dt

)= 0 .

Infinitesimal perturbationsExpand the nonlinear governing equation of the perturbation

f

(X + εξ, ε

dξ

dt

)= 0

in a Taylor series about the base solution.

f (X , 0) + εfx (X , 0) ξ + εfv (X , 0)dξ

dt= O(ε2)

To first order in ε, this can be written

Mdξ

dt= Lξ

where

M ≡ fv (X , 0)

L ≡ −fx (X , 0) .

Normal modes

Because the system is autonomous, the coefficients M and D ofthe linearized perturbation equation

Mdξ

dt= Lξ

are constant (independent of time t , though dependent on X ).Therefore the system admits normal mode solutions of the form

ξ(t) = ξest

since the derivative is

ξ(t) = s ξest

and so

sMξ = Lξ .

The eigenvalue problem

I The homogeneous linear equation

sMξ = Lξ

only has nontrivial solutions ξ 6= 0 when L− sM is singular.

I The values of s which render L− sM singular are calledeigenvalues and the associated solutions ξ, modes.

I Once the system is discretized to have a finite number n ofdegrees of freedom, ξ is an n-vector, and M , L , and L− sMare square n × n matrices.

Eigenvalues and stability

A mode with eigenvalue s evolves like

est .

In general, the eigenvalue s = σ + jω is complex.

est = e(σ+jω)t = eσtejωt

So the magnitude evolves like

|est | = |eσt | · |ejωt | = |eσt |

which only depends on the real part, σ ≡ <s ; thus:

σ ≡ <s > 0⇒ instability .

The whole spectrum of eigenvalues has to lie in the left half of thecomplex s-plane for the steady solution X to be stable.

Eigenproblems: I. The heat equation

Eigenproblems don’t only arise from linear stability analysis.Consider the one-dimensional heat equation

ρc∂T

∂t= k

∂2T

∂x2

subject toT (±1, t) = 0 .

Normal modes:

T (x , t) = X (x)est

satisfy the two-point boundary value problem

ρcsX (x) = kX ′′(x)

The heat equation

The eigenmodes of the heat equation are

X (x) =ejλnx − e−jλnx

2j= sinλnx

with

λn = nπ , n = ±1,±2, . . .

and the dispersion relation

ρcs = −kλ2 .

Thus

sn = −kλ2n/ρc < 0,∀n

and the system is stable and monotonic.

Eigenproblems: II. The vibrating string

A second example of an eigenproblem, the vibrating string:

ρ∂2y

∂t2= k

∂2y

∂x2.

Can be reduced to our canonical first-order form by introducingvelocity v as auxiliary variable.[

1 00 ρ

]∂

∂t

{yv

}=

[0 1

k ∂2

∂x20

]{yv

}

Vibrating string, cont.

Normal modes (same BCs as for heat equation):{y(x , t)v(x , t)

}=

{YV

}est sinλx

satisfy

s

[1 00 ρ

]{YV

}=

[0 1−kλ2 0

]{YV

}with characteristic equation

∣∣∣∣ s −1kλ2 sρ

∣∣∣∣ = ρs2 + kλ2 = 0

s = ±jλ√k/ρ .

All s purely imaginary, so neutrally stable and oscillatory.

Eigenproblems: III. Mass–spring–damper

mx + r x + kx = 0

Again reduce second-order equation to first-order system.[1 00 m

]d

dt

{xv

}=

[0 1−k −r

]{xv

}with characteristic equation

ms2 + rs + k = 0

and eigenvalues

s = − r

2m±√( r

2m

)2− k

m.

Stable for r > 0 , monotonic for r > 2√mk .

The Navier–Stokes equation

∂u

∂t+ u · ∇u = −1

ρ∇p + f(u) + ν∇2u

∇ · u = 0 .

Steady solution:

U · ∇U = −1

ρ∇P + f(U) + ν∇2U

∇ ·U = 0

Then consider perturbation

u ≡ U− u .

Navier–Stokes perturbation equation

Nonlinear equation for perturbation u ≡ u−U :

∂u

∂t+ u · ∇U + U · ∇u + u · ∇u =− 1

ρ∇p + f(U + u)− f(U)

+ ν∇2u

∇ · u =0 .

Linearized:

∂u

∂t+ u · ∇U + U · ∇u = −1

ρ∇p + f(U)′u + ν∇2u

∇ · u = 0 .

Steady one-dimensional base-flow

For steady one-dimensional base-flows (see Part III for 2-D):

U(x , y , z , t) = V (z)j .

Then (ignoring f for 1-D, will return to it in Parts II & III)

∂u

∂t+ w

dV

dzj + V

∂u

∂y= −1

ρ∇p + ν∇2u

∇ · u = 0 .

Normal modes (since coefficients independent of x , y , and t):

estej(kxx+kyy) .

The kx and ky are the spanwise and longitudinal wavenumbers.

Normal mode equations

su + wdV

dzj + jkyV u =− 1

ρ

{kd

dz+ jkx i + jky j

}p

+ ν

(d2

dz2− k2x − k2y

)u

j (kx u + ky v) +dw

dz=0 .

Elimination of pressure

Extract the poloidal1 part, apply

i · ∇× ≡ ∂

∂y(k·)− ∂

∂z(j·) ≡ jky (k·)− d

dz(j·)

which eliminates the pressure:{jky (k·)− d

dz(j·)}{

kd

dz+ j (kx i + ky j)

}p = j

(ky

d

dz− d

dzky

)p = 0 .

Applying it to the velocity gives the spanwise vorticity

ξ = i · ∇ × u = jky w −dv

dz.

1McBain, G. D. (2005). Plane poloidal-toroidal decomposition of doublyperiodic vector fields. The ANZIAM Journal 47

Squire’s theorem

s ξ − wd2V

dz2− k2yV w − jkyV

dv

dz+ j

dV

dzu = ν

(d2

dz2− k2x − k2y

)ξ

dw

dz= −j (kx u + ky v) .

But for one-dimensional base flows, Squire’s theorem shows thatit’s the two-dimensional (kx = u = 0) disturbances that are criticalso introduce the stream-function ψ by

u ≡ ∇× (ψi)

in terms of which

ξ = k2y ψ − ψ′′

u = 0

w = −jky ψv = ψ′

The Orr–Sommerfeld equation

cMψ = Lψ

where the eigenvalue is taken as the phase-speed

c = js/ky

and the mass and stiffness matrices are

M = k2y −D2

L =ν(k2y −D2)2

jky+ V (k2y −D2) + V ′′

whereD ≡ d/dz .

Numerical solution of the Orr–Sommerfeld equation

I discretizationI spectralI finite differenceI finite element

I generalized algebraic eigenvalue problemI shooting methods: don’t form M or LI standardize: cψ = [M−1L]ψI QZI sparse iterative methods

I generalized power method, shift-and-invert: single eigenvalueI Krylov, Arnoldi: multiple eigenvalues in a region of spectrumI libraries: ARPACK, SLEPcI false time-stepping (Tuckerman & Barkley 2000)

Shooting methods for Orr–Sommerfeld

I Shooting methods dominated in 1960s.

I Until discovery of Chebyshev-τ method2

I Persisted as ‘Riccati method’3, ‘compound matrices’4, &c.

I Recommended in Drazin & Reid’s (1981, 2004)Hydrodynamic Stability

I Doesn’t generalize to higher-dimensional base-flows.

Shooting methods are very good for nonlinear two-point boundaryvalue problems on unbounded domains; e.g., similarity solutions:

Blasius flat plate

Falkner–Skan wedge

Pohlhausen–Schmidt–Beckmann hot vertical plate

Sparrow–Gregg heated vertical plate

2Orszag 1971 JFM 503Davey 1977 J. Comp. Phys. 244Allen & Bridges 2002 Numer. Math. 92

Spectral methods for Orr–Sommerfeld

I Galerkin method used early in Russia (Gershuni 1953)I O. K., but struggled using typical basis functions.

I Superseded by methods based on orthogonal polynomials.

I Chebyshev-τ (Orszag 1971)I orthogonal collocation

I backgroundI Frazer, Duncan, & Collar (1938, Elementary Matrices)I Villadsen & Stewart (1967, Chem. Eng. Sci. 22)I Weideman & Reddy (2000, ACM TOMS 26)

I applicationsI McBain (2003, 7th Aust. Natural Convection Workshop)I McBain & Armfield (2004, ANZIAM J. 45E)I McBain & Armfield (2004, 15th AFMC)I McBain, Armfield, & Desrayaud (2007, JFM 587)I McBain, Chubb, & Armfield (2009, JCAM 224)

I Accurate but inflexible; boundary conditions finicky.

Finite differences for Orr–Sommerfeld

I Had been used very successfully by Thomas (1952).5

I Not popular.

I Actually surprisingly flexible and easy to program.

I Key: nonuniform grids.

I Quite reasonable accuracy.I See:

I Drazin & Reid (2004, § 30.2)I McBain, Armfield, & Patterson (2007–2017, unpublished)

‘Linear stability of conjugate natural convection in hot andcold fluid bodies separated by a conducting vertical wall’

I http://bitbucket.org/gdmcbain/octave

5The stability of plane Poiseuille flow. Physical Review 86

Finite elements for Orr–Sommerfeld

I Not an obvious choice as equation is fourth order.

I But so is Euler–Bernoulli’s beam equation.

I So can use Hermite elements (Mamou & Khalid 2004)6

I Excellent resultsI Advantage:

I ψ(0) = ψ′(0) = 0 are both essential boundary conditions.I Hermite elements represent each with a degree of freedom.I Differs from all ordinate-based methods:

I finite differencesI pseudospectralI cardinal basis functionsI Lagrange v. Hermite interpolation

6Intl J. Num. Meth. Fluids 44

The finite element method: weak formationGo back to eigenvalue problem for conduction of heat

ρcsu = ku′′

Weak formuation:

sρc〈v , u〉+ k〈v ′, u′〉 = [v , ku′]

Introducing basis functions φ , so that

u(z) ≈∑j

φj(z)uj

the Galerkin equation is:

sρc〈φi , φj〉uj + k〈φ′i , φ′j〉uj =

sMu − Lu = boundary terms

where the matrices are

M ≡ ρc〈φi , φj〉 and L ≡ −k〈φ′i , φ′j〉 .

Weak formulation of the Orr–Sommerfeld equation

cMψ = Lψ , in weak form:

M = k2yG0 + G1

L =ν

jky

(k4yG0 + 2k2yG1 + G2

)+ k2y 〈φi ,Vφj〉 −

⟨φi ,Vφ

′′j

⟩−⟨φi ,V

′′φj⟩

where

Gk ≡⟨dkφidzk

,dkφjdzk

⟩.

Note: M and L depend on Reynolds and wavenumbers.

Finite element method: assembly

I Standard: see, e.g.,I Becker, Carey, & Oden (1981–1986) Finite Elements, 6 vv.I Hughes (2000) The Finite Element MethodI Erm & Guermond (2004) Theory & Practice of Finite Elements

I Useful library routines:

Octave sparse

Python scipy.sparse.coo matrix

UFL (doesn’t allow 1-D Hermite elements)

Boundary conditions

I Badly applied boundary conditions mar spectra.

I Many ad hoc remedies in the literature.I A neat idea (Roy H. Stogner, libMesh mailing list):

I map away the constrained degrees of freedomI based on method for ‘hanging nodes’I Graham F. Carey (1997) Computational GridsI Carey, Stogner, libMesh all from U. Texas at AustinI given in context of finite element methods

I which already handle ‘natural’ boundary conditionsI but need special treatment of ‘essential’ boundary conditions

Essential boundary conditions, Texas style

For sMx = Lx but x constrained, say

x = Uu + Kk

then (with k = 0 for homogeneous boundary conditions)

sMUu = LUu

Render square again by projection, i.e. left multiplying by UT :

sUTMUu = UTLUu

which is back to original form:

sM′u = L′u .

Note: Preserves hermiticity & positivity.

Essential boundary conditions, Texas style: example

Lumped steady laminar flow along a duct:[+1 −1−1 +1

]1

R

{p0p1

}=

{q0q1

}with specified pressure pin at inlet but unknown outlet pressure pout{

p0p1

}=

[01

]pout +

[10

]pin

so

[01

]T [+1 −1−1 +1

] [01

]poutR

=

[01

]T{qinqout

}−[

01

]T [+1 −1−1 +1

] [10

]pinR

pout = Rqout + pin .

Generalized algebraic eigenvalue problemI. Reduction to standard form

Octave lacks routines for nonhermitian complex generalizedalgebraic eigenvalue problem

cMu + Lu = 0

so provided mass matrix is nonsingular, reduce to standard form

cu +[M−1L

]u = 0 .

Then use eig; e.g., test stability of whole spectrum with

any (imag (Rk(2) * eig (M\L)) > 0)

Alternatively, even if M is singular, shift-and-invert:

[(L− σM)−1M

]u =

(1

c − σ

)u .

Generalized algebraic eigenvalue problemII. Without reduction to standard form

I Much better not to reduce, since LAPACK, ARPACK, SLEPc,&c., all know about generalized algebraic eigenvalue problem

I even nonhermitian complex ones.

I Don’t solve for whole spectrum, just find eigenvalues withlargest real part (for s , or largest imaginary part for c).

I In Python, assuming L and M are built withscipy.sparse.coo matrix:

scipy.sparse.linalg.eigs(L.tocsc(), M=M, which=’LR’)

Tracing stability margins

I Pencil sM− L depends on Re and ky .

I Stability margin divides the Re–ky stability plane.

I Stability at given (Re, ky ) from eigenproblem.

I Trace margin using adaptive numerical continuationI McBain (2004) Skirting subsets of the plane, with application

to marginal stability curves. ANZIAM J. 45(E)I Given a stable and an unstable point.I Locate margin by one-dimensional bisection.I Test point forming an equilateral triangle.I Use it to replace stable or unstable point.I Repeat.I Adapt stable–unstable pair during bisection.

Skirting subsets of the plane (CTAC2003)

Extension to natural convection

I Orr–Sommerfeld + temperature & buoyancy

I second-order equation for temperature

I momentum & temperature equations strongly coupled

I thermal modes in addition to shear instabilities

I Gershuni (1953, Zh. Tekhn. Fiz. 23)

I Plapp (1957, J. Aero. Sci. 24)

Case studies: I. Convection in a slot (CTAC2003)

Case studies: I. Convection in a slot (CTAC2003), cont.

Case studies: I. Convection in a slot (CTAC2003), cont.

Case studies: I. Convection in a slot (CTAC2003), cont.

Case studies: II. Heated vertical wall (15AFMC )

Case studies: II. Heated vertical wall (15AFMC ), cont.

Case studies: II. Heated vertical wall (15AFMC ), cont.

Sub- and supercritical bifurcation

I For some flows, linear stability analysis gives useless results.I Couette: linear critical Re =∞I Poiseuille: Rec 2–3 times experimental value

I For others, very good:I Taylor–CouetteI Rayleigh–BenardI Blasius & Falkner–Skan boundary layersI side-heated vertical cavities & walls, as in above case studies

I The main distinction is the nonlinear behaviour of the criticalmode above and below the critical Re.

I Subcritical modes only exist below the critical point and soare nonlinearly unstable for finite amplitudes.

I Supercritical modes grow gradually as Re is increased.

Further details

papers https://www.researchgate.net/profile/

Geordie_McBain

slides https://www.slideshare.net/GeordieMcBain

code https://bitbucket.org/gdmcbain/octave

AMME BE theses:

I Chapman, C. C. 2006 Fast numerical methods for the solutionof problems in hydrodynamic stability

I Chubb, T. 2006 Solution to the Orr–Sommerfeld equationusing Green’s functions and product integration

Next

Part II.Physics of flow oscillations in the print-zone

S. G. Mallinson

Part III.Two-dimensional linear hydrodynamic stability analysis

G. D. McBain