Lebesque Measures for Transient Response of the

Navier-Stokes Equations

Kumar M. Bobba∗

University of Massachusetts, Amherst, MA 01003, USA

A thorough understanding of the response of the Navier-Stokes equations involves un-

derstanding both the transient response and the steady state response. The steady state

response of the Navier-Stokes equations has been well studied. In this work we study

the transient response of the Navier-Stokes equations linearized about plane Couette flow

under spatial and temporal varying disturbance forcing. The forcing and response are as-

sumed to belong to Lebesque spaces L2 and L∞. An analytical characterization is given for

the induced norms that characterize the response. It is shown that the L2 induced norm is

tightly bounded by the H∞ norm of the transfer function and the L∞ induced norm is up-

per bounded by the L1 norm of the impulse response operator. The structure of the worst

case disturbances and their amplification rates are computed using spectral methods with

Fourier modes in spanwise direction and Chebyshev collocation in wall normal direction.

The relevance of the present results to the channel flow experiments is discussed.

I. Introduction

Athorough understanding of the response of the Navier-Stokes equations involves understanding both

the transient response and the steady state response. The steady state response of the Navier-Stokesequations has been a subject of intense study for about a century. Here, one linearizes the Navier-Stokesequations about a given base flow and studies the spectrum of the linearized operator for different Reynoldsnumber (R) or some other non-dimensional number in the problem. The flow is said to be unstable at acertain critical Reynolds number when an eigenvalue first shows up in the open right half plane. See theclassic references1, 2, 3 for details and other references. The foundations of formulating hydrodynamic stabilityproblem was formulated some 125 years ago by Helmoltz4 and Lord Rayleigh.5 This way of formulatinghydrodynamic stability has received widespread acceptance due to the spectacular theoretical prediction ofT-S waves in Blasius boundary layer transition by Tollmien and Schlichting,6 and subsequent painstakingexperimental verification of the T-S waves by Schubauer and Skramstad7 after 20 years.

There has been a lot of mismatch between hydrodynamic stability theory predictions and experimentsin channel flows (Pouiselle, Couette, boundary layer and pipe flows, etc.) with respect to critical Reynoldsnumber at which the flow transits to turbulent state. For example, in Couette flow transition is observedin experiments any where above Reynolds number of 350, even though the flow is stable for all Reynoldsnumber according to linear infinite-dimensional proof.8 It has been observed experimentally that in openflows, transition can be postponed indefinitely if one minimizes the disturbances in the external environment.9

In fact, the experimental verification of T-S waves took 20 years after the theoretical prediction preciselyfor this reason.7 The investigators constructed a special low disturbance wind tunnel so as to observe T-Swaves. In these wall bounded flows one sees streamwise vortices and streaks,10, 11, 12, 13, 14, 15 and not T-Swaves in the natural environment. Normal mode stability analysis, on the other hand, reveals that vorticesare not the eigenfunctions of the respective linearized equations. It has been known for a long time, thatthe boundary layer streamwise vortices are the primary turbulence producing and sustaining mechanismsaway from the wall. Hence, understanding the creation of vortices and streaks from a rigorous instabilityperspective is important.

∗Assistant Professor, Mechanical and Industrial Engineering, 160 Governors Drive, AIAA Professional Member, Email:bobba@ecs.umass.edu

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That perturbations can grow transiently and decay at later times has been recognized a long time backby Orr in his seminal 1907 paper.16 For some reason, interest in this line of thought is lost very soon influid mechanics. Occasionally a paper is published here and there about transient growth of perturbationsin different forms, like resonant modes17 or algebraic growth.18, 19, 20, 21 Only recently has transient growthreceived widespread attention in fluids community with the works of.22, 23, 24, 25 Excellent reviews are givenby Schmid and Henningson,26 and Grossmann.27

The past experiments clearly indicate that transition is a strong function of the external disturbance envi-ronment. The motivation for this study stems from this observation that there are wide variety of time/spacevarying disturbance uncertainties in any fluid flow phenomena and it is natural to understand the behaviorof fluid flow under these natural uncertain conditions. For example, some of the common uncertainties intransition to turbulence in channel flows are are free stream disturbances, rough wall boundary conditions,uncertain base flow, compressibility, deterministic and stochastic forces, etc. Each of these effects enter thenominal stability equations in a particular way. In this work we focus on the disturbance uncertainty andthe response of the Navier-Stokes equations to this. Insights gained from this work can be valuable in ourunderstanding, predicting and controlling wall bounded turbulence and transition phenomena that are ofsuch immense engineering and scientific relevance.

Non-normality (i.e. non-commutation of operator and its adjoint) of the underlying governing equationsis the starting point for our work. Since the operators are non-normal, transient response has far richerstructure than when the operators are normal. This structure is yet to be fully explored. Continuing onour work in the previous papers28, 29, 30 wherein we suggested the idea of robust stability or stability underuncertainty, we study the transient dynamics of the linearized streamwise constant Navier-Stokes equationsunder time and space dependent disturbance forcing. Though the worst case response with respect to initialconditions is well studied in the literature, I believe this is the first time a study is undertaken with respectto time varying forcing that belong to Lp space with p = 2, ∞.

II. Model Equations: Incompressible Plane Couette Flow

A nice model describing the important dynamics in three dimensional Navier-Stokes is streamwise con-stant Navier-Stokes equations. This model depends on two spatial dimensions (y, z) and has three velocitycomponents (u, v, w) and as a result we call it 2D/3C model. In our previous work30 we showed theoreticallythat the non-linear 2D/3C model: is globally stable for all Reynolds number R about Couette flow, totalenergy scales like R3 and time scales like R, and R can be eliminated from the equations by a transformationthat leaves the boundary conditions invariant.

The 2D/3C model linearized about Couette flow between plane walls can be written as

∂ψ

∂t=

1

R∆−1∆2ψ − ∆−1 ∂f2

∂z+ ∆−1 ∂f3

∂y(1)

∂u

∂t= −

∂U

∂y

∂ψ

∂z+

1

R∆u+ f1 (2)

after some manipulations. Here u is the streamwise velocity, ψ is the cross-sectional stream function definedas

v ≡∂ψ

∂z, w ≡ −

∂ψ

∂y(3)

and ∆ is the two dimensional Laplacian

∆ ≡∂2

∂y2+

∂2

∂z2. (4)

U is the Couette base flow given by (1 + y)/2; f1,−f2 and −f3 are the space (depending on y and z coor-dinates) and time varying disturbance forcing in the Navier-Stokes equations. The above 2D/3C equationsare constrained by the no slip boundary conditions on the solid walls located at at y = ±1, giving

∂ψ

∂y(y = ±1, z, t) =

∂ψ

∂z(y = ±1, z, t) = 0 (5)

u(y = ±1, z, t) = 0. (6)

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III. Function Spaces and Notation

The 2D/3C equations of the previous section can be written in the operator form as

x(t) = Ax+Bw (7)

y(t) = Cx,

where x = [ψ u]t is the state vector and w = [f1 f2 f3]t

is the disturbance vector. Here t denotes transposeand ˙ denotes time derivative. Note that y(t) in the above equation is the output vector and is different fromthe coordinate axis y. The meaning should be obvious from the context. A,B and C are linear operators,while the operators A and B have their inherent meaning, the output operator C is chosen such that theEucledian 2 norm of y is the instantaneous kinetic energy of the time dependent flow.

Taking the Laplace transforms of Eq. (7) with zero initial conditions we get the frequency domaincharacterization of the system

y(s) = C(sI −A)−1Bw(s) ≡ G(s)w(s), (8)

where G(s) is called the transfer function of the system. In time domain, the solution (with zero initialconditions) can be written as a convolution between impulse response operator and disturbance input. Thatis

y(t) = g ∗ w ≡

∫ t

0

g(t− σ)w(σ)dσ, (9)

where g(t) = CeAtB is the inverse Laplace transform of the transfer function G(s).The norms of functions and operators will be denoted by ‖ ‖ with the appropriate subscript. The space

of Lebesgue integrable functions will be denoted by L with the appropriate subscript. L2 denotes the Hilbertspace of functions with the norm

‖h(t)‖L2≡

√

∫ ∞

0

‖h(t)‖22dt. (10)

L∞ denotes the Banach space of functions that have bounded

‖h(t)‖L∞≡ sup

t≥0‖h(t)‖2 (11)

norm. Here h(.) : R 7→ Cn and ‖ ‖2 denotes Eucledian 2 norm.

IV. Finite Time Response of the Navier-Stokes Equations

In this section we study the finite time response of the 2D/3C Couette flow equations when the disturbancebelongs to L2 and L∞ Lebesque spaces. For this purpose, we define two transient measures: energy normto energy norm and peak norm to peak norma. We discuss then in detail below.

A. Energy Norm to Energy Norm

Consider the linear flow system given by Eq. (7). The energy to energy norm is defined as

IEE = supw∈L2

‖y(t)‖L2

‖w(t)‖L2

. (12)

Our objective is to calculate this amplification measure, under mild assumptions. We first state the resultbelow. Assume that A is Hurwitz for all R. Then, IEE <∞ and is given by

IEE = ‖G(s)‖H∞≡ sup

ω∈R

σ[G(jω)], (13)

where G(s) is the Laplace transform of the impulse response operator and σ[H ] denotes the maximumsingular value of H .

aAnalogous results are used in robust controls literature for performance specifications in designing feedback control systems.

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The proof of the above result is described next. Let y(jω) be the time Fourier transform of y(t) andsimilarly for w(t). Then we have

y(jω) = G(jω)w(jω) (14)

for x0 = 0. Therefore, using the Parseval’s inequality we have

‖y(t)‖2L2

=1

2π

∫ ∞

−∞

w∗(jω)G∗(jω)G(jω)w(jω)dω (15)

≤ (supω∈R

σ [G(jω)])21

2π

∫ ∞

−∞

w∗(jω)w(jω)dω (16)

= ‖G‖2H∞

‖w‖2L2. (17)

Taking the square root we get that ‖G‖H∞is an upper bound for IEE .

Now we shall show that ‖G‖H∞is actually a least upper bound. Let

G(jω0) = U(jω0)Σ(jω0)V∗(jω0) (18)

be the singular value decomposition (SVD) of G(jω0), where w0 is the frequency where the maximum ofσ [G(jω)] is attained. Let v1(jω0) be the right first unit singular vector and write this as

v1(jω0) =

a1ejθ1

...

amejθm

. (19)

Pick bi such that

tan θi =−2biω0

b2i + ω20

. (20)

Construct the disturbance w as

wσ(s) =

a1b1−sb1+s...

ambm−sbm+s

a[

e−(ω−ω0)2σ/4 + e−(ω+ω0)

2σ/4]

, (21)

where

a =

(

1

1 + e−ω0σ/2

)1/2

(πσ

2)1/4. (22)

This disturbance has the property that ‖wσ(t)‖L2= 1 and as σ → ∞

a[

e−(ω−ω0)2σ/4 + e−(ω+ω0)

2σ/4]

→ π [δ(ω − ω0) + δ(ω + ω0)] . (23)

Using the above relations we get

‖yσ(t)‖2L2

=1

2π

∫ ∞

−∞

w∗σ(jω)G∗(jω)G(jω)wσ(jω)dω (24)

→ ‖G‖2H∞

, σ → ∞. (25)

B. Peak Norm to Peak Norm

Consider the linear flow system given by Eq. (7). The peak to peak norm is defined as following:

IPP = supw∈L∞

‖y(t)‖L∞

‖w(t)‖L∞

. (26)

We define the L1 norm of multi-input multi-output (MIMO) impulse response function g(t) = gij(t) as

‖g(t)‖L1=

∫ ∞

0

‖g(t)‖2→2dt, (27)

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where‖g(t)‖2→2 ≡ σ [g(t)] . (28)

We assume that g(t) ∈ L1 in this section. We will show that an upper bound for the peak to peak inducednorm is

IPP ≤ ‖g(t)‖L1. (29)

For single-input single-output (SISO) models the above bound is tight.

The proof is as following. We have

‖y(t)‖2 = ‖

∫ t

0

g(τ)w(t − τ)dτ‖2 (30)

≤

∫ t

0

‖g(τ)w(t − τ)‖2dτ ≤ ‖w(t)‖L∞

∫ t

0

‖g(τ)‖dτ. (31)

Note that g(τ) is an operator, and hence by ‖g(τ)‖ we mean the induced operator norm i.e., ‖g(τ)‖ =‖g(t)‖2→2 = σ[g(τ)]. Taking supremum with respect to t on both sides we get

IPP ≤

∫ ∞

0

‖g(τ)‖2→2dτ. (32)

One can show that the upper bound is tight for SISO systems by considering the following disturbance

w(t− τ) = sgn [g(τ)] , ∀τ, (33)

where sgn means sign of and sgn [h] = 1 if h ≥ 0, and sgn [h] = −1 if h < 0. Then ‖w‖L∞= 1 and

y(t) =

∫ t

0

g(τ)w(t − τ)dτ =

∫ t

0

|g(τ)|dτ. (34)

Therefore

‖y(t)‖L∞=

∫ ∞

0

|g(τ)|dτ = ‖g(t)‖L1. (35)

V. Computational Details

Computations are done on the 2D/3C equations linearized about Couette flow U = (1 + y)/2, by us-ing a finite-dimensional approximation of infinite-dimensional equations using spectral methods. Fouriertransforms are used in the spanwise direction and Chebyshev collocation in the wall-normal direction withChebyshev-Gauss-Lobatto points and Lagrange derivatives. The interpolating polynomials for computingthe derivatives are chosen to be Lagrange polynomials ηk(y) of degree N.

The ψ equation with boundary conditions given by Eq. (5) needs special attention. This is becausesatisfying boundary conditions is slightly complicated, as the interpolants do not satisfy the boundary con-ditions. We suitably modified the interpolants, so as to satisfy both the boundary conditions in Eq. (5). Ithas been noted31, 32 that spectral tau discretization gave spurious eigenvalues for many problems. To avoidthis, we use the pseudo-spectral approximation for Eq. (1). Here we discretize the second order differentialoperator in y (D2) using a polynomial of degree N with the boundary conditions g(±1) = 0 and fourth orderoperator (D4) using a polynomial of degree N + 2 with boundary conditions f(±1) = f ′(±1) = 0.

H∞ norm is computed by an iteration procedure using the bounded real lemma and checked using theSVD of the transfer function G. The worst case (or optimal in the sense defined previously) perturbationvectors are got from the SVD. SVD is implemented using LAPACK routines. All the computations are doneusing single processor (Pentium 3) Dell workstation running on Red Hat Linux OS.

VI. Results and Discussion

Figure 1 shows the variation of singular values of the transfer function G(s) of the linear 2D/3C model asfunction of the frequency and the spanwise wavenumber α. The figure indicates that the H∞ norm is large

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0

2000

4000

6000

σ1

0

1

2

3

4

5

α-4

-3

-2

-1

Log(ω )

X Y

Z

Figure 1. Maximum singular values of G(jω, α) at R = 1000. H∞ norm is the projection of the surface on the(x,z) plane.

even at a moderate Reynolds number like 1000. Furthermore, there is a distinctive wavenumber at whichthe induced gain peaks indicating that the flow amplifies particular wavenumber disturbances.

Figure 1 shows that the peak in the H∞ norm is achieved at ω = 0 (i.e. the low frequency limit) andα = 1.5. The first input singular vector v1 given by Eq. (19) at these parameters is plotted in figure 2as a function of the wall-normal distance. Note that v1 is composed of f1(y, z, t), f2(y, z, t) and f3(y, z, t).Actually the Fourier transform in z of the above vectors are plotted in figure 2. For notational simplificationwe retain the same symbol.

The first output singular vector u1 at the same parameter values as in figure 2 is plotted in figure 3. Theinput singular vector corresponds to the disturbance that gives the worst case amplification. The output y(t)of the flow when excited by this worst case singular vector is given by the output singular vector u1. Figures2 and 3 indicate that the size of f1 is two orders of magnitude smaller than f2 and f3 in the input vector. Onthe other hand, the size of f2 and f3 is two orders of magnitude smaller than f1 in the output vector. Thissuggests that the input disturbances are streamwise vortices and the output disturbances are streamwisestreaks in the 2D/3C equations. We speculate that a similar result holds for the three dimensional nonlinearequations.

VII. Conclusion

Developing notions to understand the transient (or finite time) stability of the Navier-Stokes equations,under a variety of disturbances that mimic the external conditions, is an important task in flow stabilitytheory. This becomes especially important from the point of view of practical applications wherein we areinterested in the finite time behavior of flow and not the infinite time (or asymptotic) behavior. In this paperwe studied theoretically and computationally the dynamics of linearized stream-wise constant Navier-Stokesequations, under external time varying deterministic disturbances belonging to L2 and L∞ function spaces.The results indicate that induced norms peak at approximately the span wise wavenumber of stream-wisestructures observed in the transiting channel flows. The huge peak in the induced norm indicates the extreme

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f

y

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

Re(f1)Re(f2)Re(f3),Im(f1),Im(f2)Im(f3)

Figure 2. First input singular vectors of G(jω, α) at R = 1000, α = 1.5 and ω = 0.

Re( f 1)

Rest

y

- 1 - 0. 75 - 0. 5 - 0. 25

- 0. 01 - 0. 005 0 0. 005 0. 01

- 0. 75

- 0. 5

- 0. 25

0

0. 25

0. 5

0. 75

Re( f 1)Re( f 2)Re( f 3) , I m( f 1) , I m( f 2)I m( f 3)

Figure 3. First output singular vectors of G(jω, α) at R = 1000, α = 1.5 and ω = 0.

sensitivity of transition to turbulence to uncertainties or unmodelled dynamics. The results validate our pointthat a full robust stability analysis is needed to study the response of Navier-Stokes equations. This may beone of the missing ingredients in the turbulence story.

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