Laminar to Turbulent Transition in Cylindrical Pipes

Transcription

1 Course I: Fluid Mechanics & Energy Conversion Laminar to Turbulent Transition in Cylindrical Pipes By, Sai Sandeep Tallam IIT Roorkee Mentors: Dr- Ing. Buelent Unsal Ms. Mina Nishi Indo German Winter Academy 2007

2 Presentation Plan Boundary Layer equations for laminar flow Onset of turbulence Primary stability theory Simple examples on stability Orr-Sommerfeld equation Reyleigh criteria of stability Origin of puffs and slugs Natural and forced transition Conclusion

3 Field Equations for flow The continuity Equation The Momentum Balance Equations in all three coordinates Vectorial representation of the momentum balance where τ τ τ xx xy xz = xy yy yz τ τ τ τ τ τ τ xz yz zz Dρ r + ρ divv = Dt Du p τ τ xx xy τ xz ρ = f x + ( + + ) Dt x x y z Dw p τ τ zx zy τ zz ρ = fz + ( + + ) Dt z x y z Dv p τ yx τ yy τ yz ρ = f y + ( + + ) Dt y x y z r Dv ρ Dt 0 r = f gradp + Divτ

4 Boundary Layer Equations for laminar flow Using the limit of Re to infinity i.e. essentially invicid flows Continuity Equation Boundary condition where derivatives of U wrt y vanish Putting the derivatives of U in place of pressure gradient Considering the steady state flow t x y ρ x y 2 u u v 1 p u + u + v = + ν 2 u v + = 0 x y y, u = U( x, t) U U 1 p + U = t x ρ x t x y t x y u 2 + u u + v v = ( U + U U ) + ν u 2 u v U x y x y 2 u v U u + = + ν 2

5 Plate Boundary layer and Blasius equation u + v = ν x y y 2 u v u 2 u = dψ dy dψ νu v= = η f f dx 2x ' ( ) u x u U η = v + = y = ϕ( η) y δ ( x) 0 dψ v = dx ψ = 2 νxu f ( η) dψ u = = U f ' ( η) dy ''' '' f + f f = Derivative of f wrt η η = 0: f=0 and df=0 η α: df = 1 0

6 Onset of Turbulence Transition in the boundary layer Boundary layer can also be either laminar or turbulent The factors on which the transition depends are Re Pressure Distribution Nature of wall( Roughness) Level of disturbance Waves namely Tollmien-Schlichting waves initiate the transition from laminar to turbulent They subsequently form three dimensional structure

7 Onset of Turbulence Transition in the boundary layer Two dimensional Tollmien-Schlichting waves are superimposed onto laminar boundary layer at indifference Re. This is Primary Stability Theory Because of secondary instabilities three dimensional and hence the Λ Structures develop (secondary stability theory) Λ Vortices are replaced by turbulent spots completing the transition

8 Onset of Turbulence Transition in the boundary layer

9 Onset of Turbulence (Stability Theory) Fundamentals of Stability Theory Laminar turbulent transition is a Stability Problem Small perturbations acted on the laminar flow At small Re Damping action of viscosity large enough to dampen these disturbances At high Re Damping action not sufficient and hence disturbance gets amplified and hence turbulence

10 Eigen value example Consider the system 1 0 x& Re x = y 2 & y 0 Re The eigen values of the matrix are negative The eigen values are and Re Re Eigen values poorly span the x direction xt () 1 t/re 0 e e yt ( ) Re Re 2 t/re

11 Stability Example Consider U t 2 μ U = 2 ρ y Let the disturbance be of the form On substitution and solving, 2π U = U0()cos( t y) λ 2 4π U = U0(0)exp γ t λ 2 The disturbance is damping with time The viscosity of the fluid eliminates the chance of turbulence in the flow Gradients promoting turbulence U = U (0)exp ( α β ) t 0 [ ]

12 Onset of Turbulence (Stability Theory) Primary Stability Theory Two methods of analysis Energy Method Method of small disturbances Assuming 2 D disturbance and incompressible flow 2 D Navier Stokes Equation Parallel Flow assumption U( y), V = W = 0; P( x, y) ' ' ' Superimposed values u(, xyt,), v(, xyt,), p(, xyt,) Resulting motion ' ' ' u = U + u, v= v, w= 0, p = P+ p

13 Onset of Turbulence (Stability Theory) Primary Stability Theory Inserting into Navier Stokes equation and eliminating the quadratic terms u u U 1 P 1 p U u u + U + v + + = ν ( + + ) t x y ρ x ρ x y x y 2 2 v v 1 P 1 p v v + U + + = ν ( + ) 2 2 t x ρ y ρ y x y u v + = 0 x y

14 Onset of Turbulence (Stability Theory) Primary Stability Theory Assuming that the basic flow itself satisfies the Navier Stokes equations, 2 2 u u U 1 p u u + U + v + = ν ( + ) 2 2 t x y ρ x x y 2 2 v v 1 p v v + U + = ν ( + ) 2 2 t x ρ y x y u v + = 0 x y On eliminating the pressure we have 2 equations and 2 unknowns

15 Onset of Turbulence (Stability Theory) Orr-Sommerfeld Equation Assume the following trial solution for the stream function ( ) ( xyt,, ) ( ye ) i α x ψ = ϕ βt α is real and β is complex β = β + iβ Real part is the frequency and imaginary part the amplification factor β i < 0 the wave is damped else amplified (unstable) r i

16 Onset of Turbulence (Stability Theory) Orr-Sommerfeld Equation Components of perturbation velocity ψ ( ) u ϕ ( ) i α x β = = y e t ψ v = = iα( y) ϕe y x Eliminating the pressure and 4 th order terms, i( α x βt). i = + α Re ( U c)( ϕ αϕ) U ϕ ( ϕ 2 αϕ αϕ) Inertial and frictional terms Starting point of stability in Laminar flows. This is called the Orr-Sommerfeld Equation

17 Orr- Sommerfeld Equation Parameters Re, α, c and r Of these the Re are generally fixed for a flow and we vary the wave length For a pair of this yields an eigen function ϕ( y) and eigen value c= cr + ici c i is the curve of neutral stability and separates the stable and unstable zone c i

18 Stable and Unstable zones The point on the curve where the Re is smallest is the Theoretical Indifference Reynolds Number

19 Reyleigh Equation c i = 0 is expected at high Re Neglecting the friction terms on the RHS of the OS equation gives inviscid perturbation diff equation called Reyleigh Equation 2 ( U c)( ϕ α ϕ) U ϕ = 0 A second order differential equation with the boundary conditions as follows y = 0; ϕ = 0; y = : ϕ = 0

20 Reyleigh Theorem Theorem 1 The first important general statement of this kind is the point of inflexion criterion. This states that the velocity profiles with point of inflection are unstable Theorem 2 A second important general statement says that in the boundary layer profiles the velocity of propagation for neutral perturbations (ci=0) is smaller than the maximum velocity of the mean flow c r < U e

21 Controversy to Reyleigh Theorem 1 It is valid only in the case where the disturbance amplified by a 2D wave is necessarily 2D After break down of the T-S wave in the 2D parallel flows the disturbance becomes 3D, a type of spiral waves which proceed along the stream direction Hence the controversy Source: On Releigh Theorem of inflectional velocity by Hua-Shu-Dou, National University of Singapore

22 Onset of Turbulence (Stability Theory) On the eigenvalues of Orr-Sommerfeld Equation ψ ( xyt,, ) = ϕ( y, α, β) e i α x βt ( ) The solution along with the boundary condition defines a characteristic equation F( α, β ) = 0 Expanding the function about any point ( α, β ) F 1 2 F F( α, β ) = F( α 0, β0) + ( α α 0) ( α 0, β 0) + ( α α 0) ( α 2 0, β0) α 2 α 2 F 1 2 F + ( β β 0) ( α 0, β 0) + ( β β0) ( α 2 0, β 0) β 2 β 2 F + ( α α 0)( β β0) +... α β Source: On the eigenvalues of the Orr-Sommerfeld equation By,M Gaster and R.Jordinson,Mathematics Department, University ofscotland,journal of Fluid Mechanics, Vol 72(1978), part 1,pp

23 Onset of Turbulence (Stability Theory) On the eigenvalues of Orr-Sommerfeld Equation Equating the above series to 0 and finding the values of β for some known values of α Let α vary in a circle as α = α 0 + Re iθ The technique is to discretize the equation and then find discrete values of the eigenvalues Other ways of finding out eigenvalues are Matrix iteration Shooting Technique Involving direct numerical integration are solutions of algebraic equations

24 Onset of Turbulence (Stability Theory) On the eigenvalues of Orr-Sommerfeld Equation

25 Onset of Turbulence (Stability Theory) On the eigenvalues of Orr-Sommerfeld Equation Concluding Remarks Rapid calculation of eignvalues by representing β in terms of α Simplification of derivatives as they are in turn expressed as series Direct solution of Orr-Sommerfeld equation proved to be tedious Contour integration of eigenvalues round a circle has proven to be good choice for discretized behavior

26 Orr- Sommerfeld Equation The Spectrum of eigenvalues Orr Sommerfeld equation has an infinite set of discrete eigenvalues { c n } and a corresponding complete set of eigenfunctions { φ ( y n )} Source: The Continuous spectrum of the Orr-Sommerfeld equation. Part 1. The spectrum and the eigenvalues,journal of Fluid Mechanics (1978), vol. 87,part 1, pp

27 Orr- Sommerfeld Equation A Trivial Example Consider the wave equation Solutions of the form u t u x 2 2 = 2 2 Here f(x) is of the form 2 d f 2 ω f 0 2 dx uxt (, ) = f( xe ) iωt + = u(0,) t = u(1,) t = 0 Infinite set of discrete eigen values and eigen vectors 0.5 ω = nπ, f ( x) = 2sin ( nπx) n n

28 Orr- Sommerfeld Equation A Trivial Example These eigen values form a complete set For an infinite domain, u(0, t) = 0, u( x, t) 0 as x If the second condition is relaxed then it forms a continnum with real omega f x 0.5 ( ; ω) = (2 π) sin ωx for ω 0 Hence by this example they form a continnum

29 Origin of Puffs and Slugs At Higher Re smooth and slightly disturbed inputs transition occurs because of flow instabilities This causes Turbulent SLUGS and occupy entire cross section At Re for (2000, 2700), for a disturbance at the inlet, the turbulent regions carried forward i.e. convected downstream at a velocity slightly smaller than the average velocity result in structures called PUFFS Puffs occur at lower Re and Slugs at Higher Re. Source: On transition in a pipe. Part I. The Origin of puffs and slugs and the flow in a turbulent slug By I. J WYGNANSKI AND F. H CHAMPAGNE,Journal of Fluid Mechanics, (1973), Vol 59, part 2, pp

30 Origin of Puffs and Slugs: Observations Slugs are caused by instability of the boundary layer to small disturbances in the inlet region Associated with laminar to turbulent transition Observed for Re>3200 Puffs, which are generated by large disturbances at the inlet Incomplete relaminarization process Observed for 2000<Re<2700 Slugs grow with axial distance and merge leading to an increase in intermittency factor and decrease in frequency

31 Origin of Puffs and Slugs: Observations Growth of Slug No growth of puffs as ULE and UTE of puffs are almost the same UTE decreases and ULE increases with Re This says Re large value UTE 0 fully turbulent For a fully turbulent flow, ULE = U and UTE = 0 and slug is of same order of magnitude as the length of pipe These measurements are taken by keeping one fixed and one moving turbulence detectors

32 Velocity of edges of a slug

33 Origin of Puffs and Slugs: Observations Entrance and origin of slugs and puffs Irrespective of the type of disturbance, the flow conditions inside a puff are the same Slugs do not originate at the entrance itself Slugs are product of transition in the developing boundary layer downstream of the entrance Boundary layer spot begins on one side of the wall and develops to comparable pipe size called the slug Breakdown of turbulence is a local phenomena and is not the same across the cross section The final stage of slugs is that they breakdown into spots

34 Origin of Puffs and Slugs: Observations Interior of Turbulent Slug Velocity Profile interior of the slug is some what similar to full turbulent flow Normalized fluctuation velocity plotted against normalized radial velocity Observed that the level of fluctuations inside a slug are greater than that in fully turbulent flow

35 The Equilibrium Puff Puffs are generated by large disturbances at inlet All puffs at same Re are equal in length Turbulent activity if puff is strongest in central zone No distinction between turbulent and non turbulent zone at the leading end of puff while that is not the case with turbulent slug Fluid might enter and leave the puff from the same interface Source: On transition in a pipe. Part II. The Equilibrium Puff, By I. J WYGNANSKI,M SOKOLOV AND D.FRIEDMAN,Journal of Fluid Mechanics, (1975), Vol 69, part 2

36 Natural Transition

37 Natural Transition Pressure Drop and Slug f 2τ w 2 = ρu 64 f lam = Re Slug starts at entrance and reaches the exit Slug tail moves through the pipe and finally leaves f = 2τ w 2 ρu f lam = 64 Re 0.25 f turb = / Re

38 Forced Transition To cause slugs to occur at low Re obstacles could be introduced at wall and pipe inlet There exists a range of Re which is critical depending on the height of obstacle Transition via puffs at lower Re can also be discussed

39 Conclusions Tollmien-Schlichting waves initiate the laminar turbulent transition Stability analysis is helpful to find the indifference Re Formation of puffs and slugs and then to the turbulent spots and finally the fully developed turbulent flow Puffs occur at lower reynolds number than the slugs Turbulence activity inside a slug is irrespective of source At same Re all Puffs have same length The boundaries of the slugs are relatively clearly defined than that of the puffs For increasing Re the slug dimensions increase resulting finally to complete turbulence

40 Thank You