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1 Multimedia files -2/13 Formulation of linear hydrodynamic stability problems Contents: 1. Governing equations 2. Parallel shear flows 3. Linearization 4. Orr-Sommerfeld and Squire equations 5. Eigenvalue problem 6. Inviscid linear stability problem 7. Destabilizing action of viscosity 8. Instability in space 9. Gasters transformation 10.Squires transformation 11.Completeness of the solutions of the Orr-Sommerfeld equation

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2 Governing equations + boundary and initial conditions - kinematic viscosity (continuity equation=mass conservation) NSE=momentum equations in Cartesian coordinates r=(x,y,z) Euler equations Streamwise (x), wall-normal (y) and spanwise (z) velocities y x z Free stream direction The term taking into account presence of stresses and responsible for presence for development of shear stresses i j k

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3 Reynolds number There are many ways to derive the expression for Reynolds number and display its significance Law of similarity: the flow about a body in simplest cases is determined by a characteristic velocity U [m/s], viscosity n [m 2 /s], and a characteristic size of the body L [m]. There is only one non- dimensional combination of these parameters, expressing the similarity of such flows: Re=UL/ Any other non-dimensional parameter can be written as a function of Re. In such a way the NS-equations can be made non-dimensional: Some of these combinations also have names, e.g., Strouhal number: St=2 fL/U, that is inverse of non-dimensional t.

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4 Nonlinear disturbance equations Let and P(r) be distributions of velocity and pressure of a known stationary solution of the Navier–Stokes equations of an incompressible fluid: with natural boundary conditions U(S) = 0 at boundaries (walls) S. (for simplicity vector sign is removed) U(x,y)

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5 Let us impose a disturbance u(r, t) and p(r,t), where r is a coordinate vector and t is time, so that the resultant motion U (r)=U(r)+u(r,t), and P (r) = P(r) + p(r, t), also satisfy the Navier–Stokes and continuity equations as well as the boundary conditions: U /t+( U ) U = P + 2 U /Re, ( U )=0. or (U+u)/t+((U+u) )(U+u)= P+ 2 (U+u)/Re, ( (U+u))=0. removing the parenthesis and separating the values (u,p) from (U,P) yields u/t+(U )u+(u )U+(u )u= p+1/R 2 u, ( u) = 0. Nonlinear disturbance equations, cont.

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6 Parallel shear flows + boundary and initial conditions (continuity equation=mass conservation) NSE=momentum equations Assume: The flow streamlines exhibit neither divergence nor convergence downstream: parallel two- dimensional (2D) flow. Local parallelity of the flow streamlines means that the streamlines diverge/converge slowly comparing with the processes of interest.

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7 Linearization Assume that and suppose that disturbances u, v, and w are small, so that the nonlinear (quadratic disturbance as u 2, uv, vw,…) terms in the NS-equations can be dropped. + boundary and initial conditions evolution equation for v equation for normal velocity:

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8 Linearization, cont. To describe the complete three-dimensional flow field, a second equation is needed (the third one is provided by the continuity equation). This is most conveniently the equation of normal vorticity, which describes the horizontal flow y x Free stream direction + initial conditions This pair of these ordinary differential equations equations provides a complete description of the evolution of an arbitrary linear disturbance. evolution equation for

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9 Orr-Sommerfeld and Squire equations The classical approach to the solution of such stability problems is the method of normal modes, consisting of a reduction of the linear initial-boundary-value problem to an eigenvalue problem. Let us suppose that the full solution can be expressed as a sum of elementary solutions (modes), which have the form Then, the evolution equations for the normal velocity and normal vorticity are reduced to The OS-equation with homogeneous b.c. constitutes an eigenvalue problem for normal velocity disturbances. In the Squire equation the right-hand term depending on the normal velocity is a driving force for the disturbances of normal vorticity. Arbitrary small complex amplitudes the Orr-Sorrmerfeld equation the Squire equation with boundary conditions at solid walls and in free stream,

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10 Examples of the eigenoscillations A duct with acoustic waves or electromagnetic microwave duct solid boundary vrvr vivi duct solid boundary solid boundary liquid boundary a boundary layer with an OS mode c U

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11 Eigenvalue problem

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12 Solution of the Orr-Sommerfeld equation OSE

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13 Solution of the Squire equation imaginary part real part

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14 Interpretation of modal results (some definitions)

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15 Analysis and Synthesis original momentum equations for disturbances linearization reduction to eigenvalue problem (spectral analysis) complete set of linearly- independent modes (waves) expansion of any small- amplitude disturbance to the set of the modes solution of initial value problem

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16 Spectral formulation of stability F =0

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17 Spectral formulation of stability, cont. Re=hU/

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18 Inviscid linear stability problem The instability waves related to the solutions of the Rayleigh equation are called Rayleigh waves. Note that Rayleigh equation in the final form is unchanged when is replaced by -. Thus, it is customary to consider >0. Also if is an eigenfunction with c for some then so is v * with eigenvalue c * for the same. Thus, to each unstable mode, there exists a corresponding stable mode (* means here complex conjugation). Considering the inviscid stability problem in time for two-dimensional (direct) waves, Rayleigh proved some important general theorems.

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19 Rayleighs inflection point criterion (first Rayleigh theorem) The inflection point is a point y s such that That is if the growth rate c i or i0, then 2 nd derivative of flow mean velocity U profile changes sign between flow boundaries. Formal multiplication of the Rayleigh equation by its complex-conjugate solution and integration by parts over y. The limits are taken [-1;+1], but this does not lead to a lack of generality. The procedure can be repeated for any other limits.

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20 Rayleighs inflection point criterion, cont. on

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21 Second Rayleigh theorem and the critical layer Rayleigh eqn. (R.E.) R.E. critical layer effects of viscosity at the wall (for c i 0). large velocity gradient

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22 Criterion of the maximum vorticity Analysis of stability of flows through the stability criteria: a stable, U 0, c stable, U=0, but U(U-U s )0; d probably unstable, U s =0 and U(U-U s )0. Here U s is the velocity at inflection point. Thus, only the velocity profiles with the inflection points associated with the maximum shear are unstable; i.e. the inequality U(U-U s )<0 should be satisfied over a certain range of y. Taking into account the first Rayleigh theorem, this is equivalent to the requirement of a relative maximum of the absolute value of the vorticity at y s for the instability to occur. It follows in particular from the criterion that the Couette flow is stable in the inviscid approach. : on

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23 Semicircle theorem stable unstable U min and U max are minimum and maximum velocity in the flow

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24 Behavior of u and v of 2D wave at finite Re in the critical layer U U inflection point Falkner-Skan basic velocity profile with backflow, H = 0.1 backflow =0.06 The streamwise velocity tends to infinity in the inviscid limit like: unless U''=0 in the critical layer.

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25 Destabilizing action of viscosity As the Reyleigh inviscid instability acts at the Reynolds number Re, it is clear that viscosity and, as a consequence, the viscous instability can destabilize the flow only at a finite Reynolds number. The physical mechanism of the viscous instability can be seen from the equation of energy balance: Considering two-dimensional disturbances in a parallel flow, the equation can be simplified to the form: re is a This term is zero without the shift

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26 Examples Different asymptotic behavior at Re. That is: The inviscid fluid may be unstable and the viscous fluid stable. The effect of viscosity is then purely stabilizing. The inviscid fluid may be stable and the viscous fluid unstable. In this case viscosity would be the cause of the instability. Results of viscous computations

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27 Disturbance source Disturbance source Instability in space The problem of boundary conditions or amplification in space. If an external signal at the entrance to the system decays whilst propagating in it, it is said that the spatial attenuation (non-transmission of a signal) takes place. Otherwise, there is a spatial amplification, and the system is called convectively unstable. In physics such systems are called sometimes amplifiers. The problem of initial conditions or stability in time. If the initial disturbance decays in time at each fixed point of space (or, at least, does not monotonically grow), the system is called stable to these disturbances. Otherwise, if the initial disturbance monotonically grows in time at a fixed point of space, the system is called absolutely unstable. In physiscs such systems are denoted sometimes as generators.

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28 evolution equation for v Difference in the evolution equations That is, we consider the development in time for the wave with given and.

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29 Difference in the evolution equations, cont. They provide quasi-linear parabolic partial differential equations in time: + Parabolic, has first order derivative in time (initial-value problem)

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30 Difference in the evolution equations, cont. A formulation of the later problem as the initial value problem is ill-posed, as the solution depends on the conditions at the downstream boundary and there are solutions propagating upstream. One has to regularize the problem by imposing additional constraints to the initial data. It was proposed to exclude all solutions propagating upstream. A possible physical explanation of this is a fast loss of the effect of the downstream boundary conditions in the bulk of the boundary layer or channel flow due to a quick decay of the upstream propagating disturbances. We will show this by considering the structure of the spectrum of the Orr-Sommerfeld and Squire equations. s: Development along x and z. For two- dimensional flows, the symmetry prescribe real, so we can consider the growth only along x. Elliptic, absence of the time derivative (b.v. problem)

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31 Eigenvalue problem The OS-equation constitutes the 4 th order polynomial (sometimes called nonlinear) eigenvalue problem in. OS SQ Note that negative i correspond to unstable disturbances, because of the factor: e i( x+ y- t) =e - i e i( r x+ y- t ).

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32 Gasters transformation

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33 Gasters transformation, cont. The prove is centered in considering the total differential of the general form of the disturbance relation about a neutral disturbance in complex plane. For strongly unstable flows, the transformation is not accurate, as it is based on a linear expansion about a neutral value. values at thу neutral curve Using it we obtain: relates small changes in a to small changes in w through the group velocity

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34 Inviscid instability in space The Rayleigh theorems is impossible to prove for complex. However, the results for neutral disturbances are equally applicable for both cases. For non-neutral cases (slightly stable or unstable) the Gasters transformation is applicable.

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35 Squires transformation It is clear that both equations are equivalent, if put

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36 Completeness of the solutions of the Orr-Sommerfeld equation For the OS and Squire equations a proof is required for the completeness of the solutions.

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37 Structure of the solutions of the Orr-Sommerfeld equation

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38 Structure of the solutions of the Orr-Sommerfeld equation (pressure waves) vrvr vivi

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39 Structure of the solutions of the Orr-Sommerfeld equation (vorticity waves) vrvr vivi downstream upstream of the source

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40 Structure of the solutions of the Orr-Sommerfeld equation (discrete waves)

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41 Further reading Betchov R. and Criminale W. O. (1967) Stability of parallel flows, NY: Academic. Drazin P. G. and Reid W. H. (1981) Hydrodynamic Stability, Cambridge University Press Gaster M. (1962) A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability, J. Fluid Mech., Vol. 14, pp Schmid P.J., Henningson D.S. (2000) Stability and transition in shear flows, Springer, p

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