sandbox/easystab/LectureNotes_SpatioTemporal.md

    Spatial and spatio-temporal stability analysis (chap. 9)

    Introduction

    Up to now, we have considered a temporal stability framework which consists of considering modal perturbation proportional to e^{i k x} and considering their possible growth/decay in time. Such perturbations represents wave-like perturbations occupying the whole domain x \in [ -\infty,\infty] and it is somewhat unphysical to consider a single wave of this kind as an initial condition.

    In this chapter we will modify this starting point in order to address two related questions :

    • what happens if a shear flow is perturbed by an initial condition localised in space ? (Initial value problem)
    • what happens if the shear flow is continuously forced by spatially localized activator immposing a sinusoidal forcing at a (real) angular frequency \omega ? (Signaling problem)

    Note : As in chapters 6-7 we restrict to 2D perturbations (\beta = 0) and still assume the base flow to be parallel.

    The signaling problem.

    Starting point

    We will first address the second question. Considering that the forcing (applied at x=0) is periodic in time e^{-\omega t}, it seems reasonable to assume that the perturbation q' of the flow will also be periodic in time. On the other hand, the perturbation can be expected to grow in space. Thus a convenient starting point is :

    \displaystyle q' = \hat{q} e^{i k x} e^{- i \omega t} ; \quad \omega \in \mathbb{R} ; k \in \mathbb{C}

    Or, written differently : q' = \hat{q} e^{i (k_r x - \omega t)} e^{-k_i x}.

    Remarks :

    • In this formalism -k_i represents the spatial growth rate in the direction x \rightarrow + \infty. A flow is thus Spatially unstable if there exists some real \omega such that k_i <0.
    • In addition, c_r = \omega / k_r represents the phase velocity of the disturbance.
    • Note that the condition ||q'||<<1 necessary for linear stability analysis cannot be expected to hold for all x as soon as the flow is spatially unstable. Necessarily, nonlinearities are dominant after a finite distance x. It is however still possible to use the linear equations locally, in some finite interval.

    Mathematical analysis

    In the spatial framework, the linearized equations (either in primitive form for \hat{ \bf q}, or in reduced form (Rayleigh or Orr-Sommerfeld) for \hat \psi) still constitute an eigenvalue problem, but the resolution now consists of solving for the eigenvalue k as function of the (real) frequency \omega and the base-flow properties.

    For temporal stability results, the growth rate does not depend upon the mean velocity U_m because of gallilean invariance. This is not the case for spatial stability results.

    Example

    For instance, for the piecewise shear layer, the solutions \omega/k = U_m \pm i \Delta U can be inverted to

    \displaystyle k = 2 \omega \frac{ U_m \mp i \Delta U}{ U_m^2 + \Delta U^2}

    Numerical resolution

    When considering k as an eigenvalue, the linear stability equations (in primitive form) can be set under the matrical form: \displaystyle k {\mathcal B} \hat{q} = {\mathcal A} \hat{q}

    In the inviscid case, the equations can be directly reduced to this form with \hat q = [ \hat{u}; \hat{v}; \hat{p}]. It is then straightforward to build a numerical program to resolve this eigenvalue problem.

    See the program KH_spatial_inviscid.m doing the resolution for the case of the tanh shear layer.

    In the viscous case, as the wavenumber k appears quadratically in the equations, it is not possible to write directly a eigenvalue problem for \hat {\bf q} = [ \hat{u}; \hat{v}; \hat{p}]. On the other hand, this can be done for \hat q = [ \hat{u}; \hat{v}; \hat{p} ; \hat{u}_1 ; \hat{v}_1] where \hat{u}_1 and \hat{v}_1 are auxiliary unknown fields defined by \hat{u}_1 = k \hat{u}; \hat{v}_1 = k \hat{v}.

    See the program KH_spatial_viscous.m to see how this this method can be implemented to solve the spatial stability problem in the viscous case.

    Typical results for shear flows

    We consider as a prototype shear flow the tanh shear layer defined as \displaystyle \bar{U}(y) = U_m + \Delta U \tanh(y) We note R = U_m / \Delta U. If 0<R<1 the velocity is always in the positive x-direction , while if R>1 the lower layer goes in the negative direction.

    Results obtained from the numerical resolution of the eigenvalue problem (KH_spatial_inviscid.m), or from model equations (Such as the one considered in thie program) observe that :

    1. If R is large, one obtains an eigenvalue noted k = k^+ (\omega,R), which is unstable in a limited range of \omega (say Im(k^+)<0 for \omega_1<\omega<\omega_2) and is associated to an eigenmode qualitatively similar to the one predicted by the temporal theory. This mode corresponds to the same instability as in temporal theory ((Kelvin-helmholtz instability, here), except that it grows with space.

    NB in the limit R \gg 1 (or U_m \gg \Delta U) one can demonstrate that the spatial eigenvalie k = k^+ (\omega) can be deduced from the temporal growth rate \omega = \Omega(k) using the Gaster transformation :

    \displaystyle k^+ (\omega) = \omega / U_m - \Delta U / U_m \Omega( \omega / U_m)

    1. As R increases, the amplification rate of the k^+ solution increases ; moreover one observes the appearance of one (or possibly several) other branch(es) called k^- for which Im(k^-)<0 for all values of \omega.

    The fact that -Im(k^-)>0 means that the amplitude of this mode grows as x increases, which seems to predict an other spatially developing instability.
    However, thanks to a mathematical analysis (explained in next section), it can be shown that this second kind of solutions actually corresponds to perturbations propagating in the upstream direction which are spatially damped as x \rightarrow - \infty.

    1. As R increases further, one reaches a particular value R=R_c for which the spatial branches coincide ; i.e. k^+(\omega_0) = k^-(\omega_0) for some real value \omega_0 of \omega.

    2. For R<R_c, one still oberves two branches of solution k(\omega) of the spatial problem but they exchange their identity. As soon as it occurs it is no longer possible to identify them as k^+ and k^- and associate them to the developement of perturbations propagating upstream and downstream, respectively.

    NB : the actual value of R_c is 1.319 for the tanh shear layer (Huerre & Monkewitz, 1988), and R_c = 1 for the simple model considered in exercice 1.

    The initial value problem.

    Examining the failure of the spatial problem

    We have seen that for R<R_c the spatial “signaling” problem admits a well-defined solution both for x>0 (corresponding to k^+ solutions) and x<0 (corresponding to k^- solutions), while for R>R_c it leads to results for which we are not able to give a physical interpretation. What happens in such cases ???

    To understand this, reconsider the starting point of the spatial stability analysis. We have supposed that both the localised forcing at x=0 and the perturbation of flow are proportional to e^{i \omega t}. This corresponds to the permanent response, namely the response to a forcing exists for all instants starting from t=- \infty. In reality, the forcing starts at a given instant (say t=0). Then the perturbation is the sum of the permanent response and a transient response. Thus the spatial stability results are only relevant to the cases where the transient regime decays.

    Mathematical formulation of the initial value problem

    Let us go back to the initial value problem. If the initial condition is square-integrable, we can always perform a Fourier transform in x and write the solution as follows: \displaystyle q'(x,y,t) = \int_{-\infty}^{+\infty} \tilde{q}(k,y,t) e^{i k x } \, dk

    for each value of k, the evolution is given by \displaystyle \frac{\partial}{\partial t} {\mathcal B} \, \tilde{q} = {\mathcal A}_k \, \tilde{q}

    and the we have seen that the solution can be projected on the basis of eigenmodes (including discrete and continuous ones), namely : \displaystyle \tilde{q}(k,y,t) = \sum_{\omega_n \in Sp^d(k)} c_n(k) \, \hat{q}_n(k,y) \, e^{-i \omega_n(k) t} + \int_{Sp^c} c(k,\omega) \hat{q}_\omega(k,y) e^{-i \omega t} where c_n(k) and c(k,\omega) can be obtained from the projection of the initial condition q'(x,y,t=0).

    To simplify, we keep only the leading eigenmode (n=1) of the discrete spectrum and discard the continuous spectrum, namely \tilde{q}(k,y) \equiv c(k) \hat{q}(k,y) e^{-i \omega(k) t}

    Reintroducing this solution in the Fourrier transform leads to

    \displaystyle q'(x,y,t) = \int_{-\infty}^{+\infty} c(k) \hat{q}(k,y) e^{i k x} e^{-i \omega(k) t} \, dk

    where c(k) can be obtained from the x-Fourier transform of the initial condition q'(x,y,t=0).

    Asymptotic behavior

    The asymptotic behaviour of the previous expression can be predicted using the stationnary phase theorem

    (Applicable if Im ( \omega(k) ) \leq 0 as k \rightarrow \pm \infty (k \in \mathbb R), which physically means that the short-wave perturbations are not amplified).

    Theorem (stationnary phase)

    1. If there exists a (complex) k_0 such that \partial \omega/\partial k = 0 (corresponding to a saddle point of the complex function \omega(k)), and if there exists a steepest descent path in the complex k-plane linking -\infty to +\infty and passing through this saddle point; then \displaystyle q'(x,y,t) = \sqrt{\frac{ 2 \pi}{ i \omega''(k_0) t} } c(k_0)\hat{q}(k_0,y) e^{ i k_0 x} e^{ - i \omega(k_0) t} \mathrm{ as } \quad t \rightarrow +\infty \quad (\mathrm{ with \, fixed \, } x)
    2. Otherwise \displaystyle q'(x,y,t) = o(t^{-1/2}) \qquad \mathbf{ as } \quad t \rightarrow +\infty \quad (\mathrm{ with \, fixed \, } x)

    The demonstration of the theorem relies on the fact that the argument in the integral expession of q'(x,y,t) is an analytical fonction of the complex variable k. Hence, a theorem of complex anaysis states that the contour of integration from k=-\infty to k=+\infty can be arbitrarily deformed (provided it does not cross singularities of the function). Taking the aformentionned steepest descent path contour leads to the result.

    Interpretation : convective/absolute instabilities

    This results leads to the following conclusion :

    1. If the complex frequency \omega_0 = \omega(k_0) corresponding to the saddle point verifies Im( \omega_0 <0) (or if there is no such saddle point), then the perturbation will decay with time at every fixed position x. The instability is said to be convective. In this case the transient response decays and it is allowed to use spatial stability analysis to study the permanent response to a harmonic forcing.

    2. If the complex frequency \omega_0 = \omega(k_0) corresponding to the saddle point verifies Im( \omega_0 >0) , then the perturbation will grow with time at every fixed position x. The instability is said to be absolute. In this case the transient response always dominate and the spatial stability is not relevant.

    Asymptotic behaviour in a moving frame

    The notion of a convectively unstable flow may sound paradoxal: this case implies that there exists a (real) value of k such as perturbation proportional to e^{ikx} is exponentially amplified; BUT on the other hand at a given x the response to ANY spatially localized perturbation decays.

    To clarify this point let us reconduct the stationary phase argument in a frame of reference moving at a velocity V. Writing x=Vt+x' allows to re-write the solution as \displaystyle q'(x,y,t) = \int_{-\infty}^{+\infty} c(k) \hat{q}(k,y) e^{i k x'} e^{-i (\omega(k)-kV) t} \, dk

    Letting t \rightarrow \infty BUT keeping x' fixed and applying again the stationary phase therorem leads to:

    \displaystyle q'(x,y,t) = \sqrt{\frac{ 2 \pi}{ i \omega''(k_V) t} } c(k_V)\hat{q}(k_V,y) e^{i k_v x'} e^{ - i\omega_V t} \mathrm{ as } \quad t \rightarrow +\infty \quad (\mathrm{ with \, fixed \, } x'=x-Vt)

    where k_V is the solution of \partial \omega / \partial k = V and \omega_V = \omega(k_V) - k_V V the corresponding frequency (in the moving frame). If Im(\omega_V) >0, then the perturbation grows as x,t \rightarrow \infty along the ray defined by x/t = V.

    We can then define the interval [V_1,V_2] corresponding to velocities of rays along which the perturbation grow. The rays with velocities V_1 and V_2 can be interpreted as the upstream an downstream fronts of the unstable wavepacket (region in the x-t plane where the perturbation will grow).

    1. For an absolutely unstable case, the interval [V_1,V_2] comprises V=0, so an initially localized perturbation will tend to occupy a region in the x-t plane expanding both upstream (V_1<0) and dowstream (V_2>0).

    2. On the other hand, in a convectively unstable case, an initially localized perturbation will give rise to an unstable wavepacket which will be advected in the positive direction. At any fixed x, the perturbation will become large as the unstable wavepacket crosses the x location, but will eventually decay.

    Reexamining the spatial approach

    We have seen that there exists a value R=R_c at which there is a coincidence of the spatial branches ; namely there exists two solutions k^+(\omega), k^-(\omega) such as k^+(\omega_0) = k^-(\omega_0) for some real \omega_0. This coincidence of two branches precisely corresponds to a double root of the function \omega(k). The existence of a double root precisely corresponds to the condition \partial \omega / \partial k =0.

    In complex analysis, a coincidence of two branches of solutions when a parameter R crosses a value R_c is always associated to an exchance of identities of the two branches, which reconnect differently for R<R_c and R>R_c, respectively. We thus see that the transition between the situation with distinct k^+ and k^- branches for R<R_c and the situation where the branches is completely explained by the existence of a saddle point along the real \omega-axis for R=R_c.

    Exercices for lecture 9

    Exercice 1 Absolute/convective transition for a model equation.

    We consider a simple model characterised by the following dispersion relation :

    \displaystyle D(\omega,k) = (\omega - k U_m) - i \Delta U k (1-k) = 0

    1. Considering temporal stability analysis, compute \omega (complex) as function of k (real). Plot the amplification rate \omega_i as function of k/k_c. Justify that the present model can be used as an aproximation for the stability properties of a finite-thickness shear layer.

    2. Find the value k_0 corresponding to the saddle point of the relation and deduce the corresponding absolute frequency \omega_0. Show that \displaystyle \omega_0 = \frac{\Delta U}{2} + \frac{ i ( (\Delta U)^2-U_m^2) }{4 \Delta U}

    3. Deduce ranges of R= \Delta U/U_m corresponding respectively to absolute and convective instabilities. (it is admitted that the saddle point verifies the “steepest descent path” property)

    4. Compute the spatial stability branches k(\omega) of this model and plot them in the convective and absolute cases (you may start from this program and adapt it).

    Exercice 2 Numerical resolution of the spatial stability problem.

    We consider modal perturbations of a parallel shear flow under the form :

    \displaystyle [u,v,p] = [ \bar{U}(y) ,0, 0 ] + [ \hat{u}, \hat{v}, \hat{p}] e^{i k x - i \omega t}

    1. Considering non-viscous theory, show that the linear stability problem can be set under the form of a generalized eigenvalue problem:

    \displaystyle k {\mathcal B} \hat{q} = {\mathcal A} \hat{q} \qquad \mathrm{with} \quad \hat q = [ \hat{u}; \hat{v}; \hat{p}] and give the expression of the matricial operators \mathcal A and \mathcal B.

    (Solution : see the program KH_spatial_inviscid.m)

    1. In the viscous case, show that the problem can be reduced to

    \displaystyle k {\mathcal B}^v \hat{q}^v = {\mathcal A}^v \hat{q}^v \qquad \mathrm{with} \quad \hat q = [ \hat{u}; \hat{v}; \hat{p} ; \hat{u}_1 ; \hat{v}_1] \qquad \mathrm{ ; } \quad \hat{u}_1 = k \hat{u}; \quad \hat{v}_1 = k \hat{v}

    Give the expression of the matricial operators.

    (Solution : see the program KH_spatial_viscous.m)

    Recommended lectures

    For a more complete discussion of the spatio-temporal aspects of instabilities, the recommended reference is the lecture of Huerre & Rossi (1998), which is actually a chapter of a book edited by Godrèche & Manneville (1998).