Presentation on theme: "Formulation of linear hydrodynamic stability problems"— Presentation transcript:
1Formulation of linear hydrodynamic stability problems Multimedia files -2/13Formulation of linear hydrodynamic stability problemsContents:Governing equationsParallel shear flowsLinearizationOrr-Sommerfeld and Squire equationsEigenvalue problemInviscid linear stability problemDestabilizing action of viscosityInstability in spaceGaster’s transformationSquire’s transformationCompleteness of the solutions of the Orr-Sommerfeld equation
2Governing equations in Cartesian coordinates r=(x,y,z) Euler equationsNSE=momentum equationsThe term taking into account presence of stresses and responsible for presence for development of shear stresses(continuity equation=mass conservation)+ boundary and initial conditionsStreamwise (x), wall-normal (y) and spanwise (z) velocitiesin Cartesian coordinates r=(x,y,z)yFree streamdirectionjkixz- kinematic viscosity
3Reynolds numberThere are many ways to derive the expression for Reynolds number and display its significanceLaw of similarity: the flow about a body in simplest cases is determined by a characteristic velocity U [m/s], viscosity n [m2/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/ nAny other non-dimensional parameter can be written as a function of Re.In such a way the NS-equations can be made non-dimensional:In fluid mechanics, the Reynolds number Re is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions. The concept was introduced by George Gabriel Stokes in 1851, but the Reynolds number is named after Osborne Reynolds (1842–1912), who popularized its use in 1883.Reynolds numbers frequently arise when performing dimensional analysis of fluid dynamics problems, and as such can be used to determine dynamic similitude between different experimental cases. They are also used to characterize different flow regimes, such as laminar or turbulent flow: laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion, while turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities.Some of these combinations also have names, e.g., Strouhal number: St=2pfL/U, that is inverse of non-dimensional t.
4Nonlinear 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)
5Nonlinear disturbance equations, cont. 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 motionU(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∇2u,(∇u) = 0.
6Parallel shear flows NSE=momentum equations (continuity equation=mass conservation)+ boundary and initial conditionsThe flow streamlines exhibit neither divergence nor convergence downstream: “parallel two-dimensional (2D) flow”.Assume:Local parallelity of the flow streamlines means that the streamlines diverge/converge slowly comparing with the processes of interest.
7LinearizationAssume thatand suppose that disturbances u, v, and w are small, so that the nonlinear (quadratic disturbance as u2, uv, vw,…) terms in the NS-equations can be dropped.+ boundary and initial conditionsequation for normal velocity:evolutionequation for v
8Linearization, 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” flowyFree streamdirectionxevolutionequation for + initial conditionsThis pair of these ordinary differential equations equations provides a complete description of the evolution of an arbitrary linear disturbance.
9Orr-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 formArbitrary ‘small’ complex amplitudesThen, the evolution equations for the normal velocity and normal vorticity are reduced tothe Orr-Sorrmerfeldequationthe Squire equationwith boundary conditionsat solid walls and in free stream,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.
10Examples of the eigenoscillations duct solid boundarycduct solid boundaryA duct with acoustic waves or electromagnetic microwave“liquid boundary”Ua boundary layer with an OS modevrsolid boundaryvi
13Solution of the Squire equation imaginary partreal part
14Interpretation of modal results (some definitions) bkg
15Analysis and Synthesis original momentum equations for disturbanceslinearizationsolution of initial value problemreduction to eigenvalue problem(spectral analysis)expansion of any small-amplitude disturbance to the set of the modescomplete set of linearly-independent modes (waves)
17Spectral formulation of stability, cont. Re=hU/n
18Inviscid 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 a is replaced by -a. Thus, it is customary to consider a>0. Also if is an eigenfunction with c for some a then so is v* with eigenvalue c* for the same a. 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.
19Rayleigh’s inflection point criterion (first Rayleigh theorem) The inflection point is a point ys such thatThat is if the growth rate ci or wi≠0, then 2nd 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.
21Second Rayleigh theorem and the critical layer Rayleigh eqn. (R.E.)R.E.(for ci≠0).criticallayerlarge velocitygradienteffects of viscosityat the wall
22Criterion of the maximum vorticity :onHere Us is the velocity at inflection point.Thus, only the velocity profiles with the inflection points associated with the maximum shear are unstable; i.e. theinequality U”(U-Us)<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 ys for the instability to occur. It follows in particular from the criterion that the Couette flow is stable in the inviscid approach.Analysis of stability of flows through the stability criteria: a stable, U’’<0; b stable, U’’>0, c stable, U’’=0, but U’’(U-Us)≥0; d probably unstable, U’’s=0 and U’(U-Us)≤0.
23Semicircle theoremUmin and Umax are minimum and maximum velocity in the flowstableunstable
24Behavior of u and v of 2D wave at finite Re in the critical layer Falkner-Skan basic velocity profilewith backflow, bH=‒0.1UU”a=0.06inflection pointThe streamwise velocity tends toinfinity in the inviscid limit like:backflowunless U''=0 in the critical layer.
25Destabilizing 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 aThis term is zerowithout the shift
26Different asymptotic behavior at Re→∞. ExamplesResults of viscous computationsDifferent asymptotic behavior at Re→∞.That is:The inviscid fluid may be unstable and the viscous fluid stable. The effectof viscosity is then purely stabilizing.The inviscid fluid may be stable and the viscous fluid unstable. In thiscase viscosity would be the cause of the instability.
27Instability in space Disturbance source ’ 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”.Disturbance sourceThe 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”.
28Difference in the evolution equations equation for vThat is, we consider thedevelopment in time for the wave with given a and b.
29Difference 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)
30Difference in the evolution equations, cont. Development along x and z. For two-dimensional flows, the symmetry prescribe b real, so we can consider the growth only along x.Elliptic, absence of the time derivative (b.v. problem)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.
31ei(ax+by-wt)=e-aiei(arx+by-wt). Eigenvalue problemOSSQNote that negative ai correspond to unstable disturbances, because of the factor:ei(ax+by-wt)=e-aiei(arx+by-wt).The OS-equation constitutes the 4th order polynomial (sometimes called “nonlinear”) eigenvalue problem in a.
33Gaster’s 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.Using it we obtain:values at thу neutral curverelates small changes in a to small changes in w through the group velocityFor strongly unstable flows, the transformation is not accurate, as it is based on a linear expansion about a neutral value.
34Inviscid instability in space The Rayleigh theorems is impossible to prove for complex a.However, the results for neutral disturbances are equally applicable for both cases.For non-neutral cases (slightly stable or unstable) the Gaster’s transformation is applicable.
35Squire’s transformation It is clear that both equations are equivalent, if put
36Completeness of the solutions of the Orr-Sommerfeld equation For the OS and Squire equations a proof is required for the completeness of the solutions.
37Structure of the solutions of the Orr-Sommerfeld equation
38Structure of the solutions of the Orr-Sommerfeld equation (pressure waves) vrvi
39Structure of the solutions of the Orr-Sommerfeld equation (vorticity waves) upstream of the sourcedownstreamvivr
40Structure of the solutions of the Orr-Sommerfeld equation (discrete waves)
41Further readingBetchov R. and Criminale W. O. (1967) Stability of parallel flows, NY: Academic.Drazin P. G. and Reid W. H. (1981) Hydrodynamic Stability, Cambridge University PressGaster M. (1962) A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability, J. Fluid Mech., Vol. 14, pp. 222‒224.Schmid P.J., Henningson D.S. (2000) Stability and transition in shear flows, Springer, p. 1‒60.