sandbox/easystab/LectureNotes_StabilityOfParallelFlows.md

    Linear stability analysis of parallel flows : General formalism

    The objective of this section is to expose the general formalism used in the stability analysis of parallel flows, hence forming the starting points of chapters 6,7,8,9.

    Hypotheses

    • The mean flow is parallel : \vec{u} = \bar{U}(y) \vec{e}_x for y \in [y_1,y_2] ; y_1 and y_2 can be finite or infinite.

    (examples : shear layers, wakes, jets, boundary layers…)

    • Incompressible, constant density \rho \equiv 1.

    • Viscosity \nu \equiv Re^{-1} (using convenient nodimensionalization).

    Flow expansion

    We expand the flow as base flow plus pertubations

    \displaystyle {\left[ \begin{array}{c} u \\ v \\ w \\ p \end{array} \right]} \,= \, {\left[ \begin{array}{c} \bar{U}(y) \\ 0 \\ 0 \\ \bar{P} \end{array} \right]} \quad + \quad {\left[ \begin{array}{c} u'(x,y,z,t) \\ v'(x,y,z,t) \\ w'(x,y,z,t) \\ p'(x,y,z,t) \end{array} \right]}

    Which we write symbolically q = q_0 + q', introducing the the state vector notation q = [u;v;w;p]

    Here \bar{P} is either a constant (if gravity is neglected) or a hydrostatic field.

    Perturbation equations

    Injecting this ansatz in the Navier-Stokes equations and linearizing lead to the following set of equations:

    \displaystyle \left\{ \begin{array}{rcl} \partial_t u' + \bar{U} \partial_x u' + v' \partial_y \bar{U} &=& - \partial_x p' + Re^{-1} ( \partial_x^2 + \partial_y^2 + \partial_z^2) u' \\ \partial_t v' + \bar{U} \partial_x v' &=& - \partial_y p' + Re^{-1} ( \partial_x^2 + \partial_y^2 + \partial_z^2) v' \\ \partial_t w' + \bar{U} \partial_x w' &=& - \partial_z p' + Re^{-1} ( \partial_x^2 + \partial_y^2 + \partial_z^2) w' \\ \partial_x u' + \partial_y v' + \partial_z w' &=& 0 \end{array} \right.

    Notations and physical meaning

    Owing to the invariance with respect to x and z directions, the perturbation can be considered in modal form. We introduce k and \beta the axial and transverse wavenumbers. On the other hand at this stage we keep the temporal dependance.

    Hence : \displaystyle q'(x,y,z,t) = \Re [\tilde{q}(y,t) e^{i k x + i \beta z} ] (In the sequel the symbol \Re for the real part will be ommitted ; it is understood that only the real part of any complex expression is relevant).

    Note that when both k and \beta are nonzero (but real) this modal expression describes an oblique wave with wavefronts oriented at an angle \Theta = atan( \beta / k) with respect to the axial directions.

    The case \beta=0 corresponds to transverse waves (chapters 6 and 7), and in this case the velocity component w' can be dropped.

    The case k=0 corresponds to axially aligned structures and is relevant to describe axial streaks and vortices (see chapter 8).

    The case where k is a complex number is relevant to describe spatial of spatiotemporal growth of perturbations and will be considered in chapter 9.

    Equations

    When injecting into the starting equations, all axial derivatires \partial_x are replaced by ik and transverse derivatives \partial_z are replaced by i\beta. The linear equations for the perturbation can be then written as a linear algebraic problem as follows: \displaystyle \frac{\partial}{\partial t} {\mathcal B} \, \tilde{q} = {\mathcal A} \, \tilde{q}

    with \displaystyle {\mathcal B} = \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right]

    \displaystyle {\mathcal A} = \left[ \begin{array}{cccc} -i k \bar{U} + Re^{-1} ( \partial_y^2 - k^2 - \beta^2) & - \partial_y \bar{U} & 0 & - i k \\ 0 & -i k \bar{U} + Re^{-1} ( \partial_y^2 - k^2 - \beta^2) & 0 & - \partial_y \\ 0 & 0 & -i k \bar{U} + Re^{-1} ( \partial_y^2 - k^2 - \beta^2) & - i \beta \\ i k & \partial_y & i \beta & 0 \end{array} \right]

    Eigenmode analysis consists of searching for modal perturbations in space and time, i.e: \displaystyle q'(x,y,z,t) = \Re [\hat{q}(y) e^{i k x + i \beta z- i\omega t} ]

    This leads to the matricial eigenvalue problem:

    \displaystyle - i \omega {\mathcal B} \, \hat{q} = {\mathcal A} \, \hat{q}

    Note that the notation c = \omega/k (where c is complex) is often used in the litterature, hence the eigenvalue problem can also be written \left[ {\mathcal A} + i k c {\mathcal B} \right] \, \hat{q} = 0.

    Inviscid temporal stability analysis of parallel flows - Kelvin-Helmholtz instability (chap. 6)

    This chapter has been moved here

    Viscous temporal stability analysis of parallel flows - Tollmien-Schlishting instability (chap. 7)

    This chapter has moved here

    3D and transient growth mechanism in shear flows (chap. 8)

    This chapter has moved here

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

    This chapter has moved here

    Back to main page