Fast waves during transient flow in an asymmetric channel

Transcription

1 Int. Conference on Boundar and Interior Laers BAIL 6 G. Lube, G. Rapin (Eds) c Universit of Göttingen, German, 6. Introduction Fast waves during transient flow in an asmmetric channel Dick Kachuma & Ian Sobe Oford Universit Computing Laborator Wolfson Building, Parks Road, Oford UK OX 3QD dick.kachuma@comlab.o.ac.uk ian.sobe@comlab.o.ac.uk The use of stepped channels together with unstead laminar flow provides a powerful miing mechanism that is particularl applicable to processes where the fluid contains delicate elements, for eample, applications involving mass transfer in blood or in a cell culture. In such channel flows there are parameter regimes where the flow is described b the two-dimensional unstead Navier Stokes equations. In [], it was shown both eperimentall and numericall that a standing wave of separated regions develops behind a channel step during oscillator flow and the resulting flow was called a vorte wave. Included in the eperimental observations were vorte waves of etreme longitudinal etent and it was conjectured that the wave formed was, under the correct parameter conditions, virtuall undamped in the streamwise direction. We have undertaken calculations in a slightl different parameter region and find that a sequence of two events occurs: one is the formation of a vorte wave of finite etent (tpicall 3-5 vortices alternating on the two walls behind the step), the second is a subsequent rapidl propagating wave of regular but slightl smaller vortices. The speed of propagation of this second wave is such that its resolution would have been beond that of the apparatus used in []. In describing these waves we shall refer to the vorte wave as a V-wave and the second wave as a KH-wave. An eample of the waves is illustrated in figure where streamlines of a flow are shown for one time in a ccle. V-wave KH-wave Figure : Instantaneous streamlines In order to understand the genesis of KH-waves we have undertaken a stud of starting flows where the fluid is accelerated from rest to either stead channel flu or a flu with an oscillator component. The results presented below are for pure oscillator flow. We test two hpotheses: one that the KH-wave results from the evolution of an inviscid rotational core flow that is described b an evolutionar linearised Kortweg-de Vries (KdV) equation. That this might be a plausible hpothesis comes from results in [3] which show the evolution of waves in solutions of an evolutionar KdV equation. The second hpothesis is that the KH-wave results from an Orr-Sommerfeld tpe instabilit of nearl parallel but non-poiseuille like flow. This has lead us to stud stabilit of a base flow u () = (3/)( σ)( ),, where σ = is Poiseuille flow and σ > indicates reverse flow near one wall. The paper is divided into four sections. We briefl describe the numerical solution of the unstead Navier Stokes equations and illustrate the development of a KH-wave. We integrate

2 velocities in time to obtain particle paths and use these to help interpret the development of the flows. We describe solutions of an evolutionar linearised KdV equation and their interpretation. We consider solutions of an Orr-Sommerfeld equation for a parallel flow with a reverse flow region and investigate unstable solutions to that sstem. While much work is still in progress, the results we have indicate that it is unlikel that the KH-wave is the result of evolution of an inviscid rotational core flow and instead, that it is more likel the result of a linear instabilit mechanism described b an Orr Sommerfeld equation but with growth rates that are orders of magnitude greater than those for disturbances to smmetric Poiseuille flow and with instabilit occuring at relativel low Renolds number. The compleit of unstead flows calculated from the full Navier Stokes equations is remarkable and it is likel that flows in other parameter regimes ma be dominated b entirel different mechanisms.. Numerical solution of the unstead Navier Stokes equations We consider incompressible viscous flow in a two dimensional periodic channel with inlet width h and a sudden epansion to a width h. We take h as a length scale, (,) as non-dimensional coordinates with as the channel streamwise direction and fluid kinematic viscosit ν. We let L denote the non-dimensional length of one section of the channel and L h the non-dimensional length of the epansion. We show such a channel in figure which has L = 96 and L h = 84. It is such large values of L and L h that allow enough room for the fast waves to develop Figure : Schematic of two-dimensional channel If the flu per unit width of channel is Q(ˆt/T s ) where ˆt is dimensional time and T s a characteristic timescale imposed b the oscillation, define non-dimensional time t = ˆt/T s. Assuming that the peak flu is Q ma so that for some q(t), Q = Q ma q(t), define a velocit scale U = Q ma /h, a Renolds number Re and a Strouhal number St as Re = U h ν = Q ma, St = h = ν U T s h Q ma T s. For oscillator flow we have q(t) = sin πt, while if there is a mean flow Q (tpicall < ), the ratio of mean forward flu to peak forward flu, a more general form of q(t) is q(t) = Q + ( Q)sin πt with Q = impling stead flow. Since the flow field is two-dimensional we use the streamfunction-vorticit formulation, so that the velocities in the (,) directions are u = (u,v), respectivel. We define a streamfunction, ψ, and vorticit, ω, b u = ψ, v = ψ, ω = u + v = ψ. () The Navier Stokes momentum equations reduce to St ω t + (u )ω = Re ω. () The boundar conditions are ψ = q(t) on the lower wall, ψ = q(t) on the upper wall, and ψ n = on both walls, where n is normal to the walls. We impose a periodicit condition on

3 (a) (b) (c) (d) (e) (f) (i) Re = (ii) Re = 6 Figure 3: Instantaneous streamlines showing the development of a vorte wave and subsequent KH-waves. The flow is at St =.5 and Re = (i) and Re = 6 (ii) at times t =. (a), t =. (b), t =. (c), t =.3 (d), t =.4 (e) and t =.5 (f). Note that the channel spect ratio is highl distorted with length L = 96 and maimum width. both ψ and ω in the streamwise direction, ψ(,, t) = ψ( + L,, t), ω(,, t) = ω( + L,, t). The equations are integrated in time b using a semi-implicit Crank Nicolson method with standard central differences used for spatial derivatives. We show in figure 3, the development of vorte waves and the fast waves b plotting the streamlines of the computed fluid flow for Renolds numbers (i) and 6 (ii) and Strouhal number St =.5 over half a period of oscillation. At t =. (a), the vortices from the preceding half ccle are still present in the channel. As the flow is accelerated to t =. (b), these effects are eroded and at t =. (c), long wavelength vorte waves, characterised b 3 or 4 counter-rotating vortices are seen to develop. When the flow starts to decelerate at t =.3, (d) the flow at Re = 6 starts to shed shorter wavelength fast-moving waves which travel far downstream (e). The flow at Re = however, does not possess this structure and onl develops the long wavelength vorte waves. At t =.5, (f) the flu returns to zero and the flow is a reflection of that at t =. (a). In subsequent half ccles, the patterns reverse and the sequence of vorte wave formation and fast wave shedding repeats. In the case Re = 6, St =.5, the Navier Stokes solutions suggest that the wavelength of the KH-wave is 3-5 non-dimensional units and the frequenc near times that of the imposed oscillation. 3. Particle paths In this section we investigate the movement of particles in oscillator flow in a channel. The time dependent solutions of the Navier Stokes equations developed in the preceding section, will be used to trace the trajectories of a cloud of particles which is initiall uniforml distributed in one section of the periodic channel. We denote b (n), (n) the position of particle n and hence the equations of motion associated with the particle are (n) St t (n) = u((n), (n), t), St t 3 = v((n), (n), t), (3)

4 where u and v are the time dependent velocit fields of the fluid given b the solution of the Navier Stokes equations. This assumes that the particles move in the fluid without affecting the underling flow and that the move as fluid particles. We denote b Np the total number of particles and we have performed the computations for three values of Np, Np = 88, Np = 5 and Np = 468. Our calculations show that the results obtained to not var significantl with the number of particles. (a) (b) (c) (d) (e) (f) (i) Re = (ii) Re = 6 Figure 4: Particle trajectories for flows at Re = (i) and Re = 6 (ii) at Strouhal number St =.5. The particles are initiall uniforml distributed at t = (a) and are subsequentl shown at t =. (b), t =. (c), t =.3 (d), and t =.4 (e). We show in figure 4, the motion on particles. The figure shows a scatter plot of the positions of the particles at different times during one ccle of oscillation. The figure shows particles for the Renolds numbers (i) and 6 (ii) and Strouhal number St =.5 with the particles are coloured black or gra depending on the half of the channel in which the are initiall positioned. The diagram illustrates the increase in transverse miing at the higher Renolds number. As a measure of the scatter of the particles we compute the variance σ of the longitudinal position Np X (i (t) (t)), σ (t) = Np i= PNp where (t) = Np i= i (t) is the mean of the distribution. We can therefore approimate the instantaneous dispersion coefficient [4] D(t) = dσ dt and integrating this over one ccle of oscillation gives a ccle dispersion coefficient Z D(t)dt = σ K= ccle ccle Figure 5 (i) shows the instantaneous variance σ of the longitudinal position at St =.5 for Re = and Re = 6. This shows an overall linear growth of the variance over time with oscillations at the same frequenc as the underling flow. The dipersion coefficient over one ccle K is shown in figure 5 (ii) as a function of the Renolds number at St =.5. This shows that the dispersion increases with Re until Re > 5 when it begins to decline thus revealing that a rapid increase in miing occurs when the KH-wave forms. 4

5 σ 5 K (i) t Re (ii) Figure 5: (i) Instantaneous variance σ (t) of the longitudinal position of particles at St =.5 and Re = (solid) and Re = 6 (dashed). (ii) Ccle dispersion coefficient K as a function of Re at St =.5 computed over the interval t [,]. 4. Inviscid Theor We develop, in this section, an inviscid theor for the generation of the vorte wave. The work described here follows closel, the theor developed b [] and [3]. We consider a smooth-walled channel in which the size of indentations is small compared to the width of the channel. We suppose that the lower wall of the channel is fied at = and that the upper wall is defined b = + ǫf() where ǫ and F is a smooth function. A long wavelength approimation is made for the vorte wave and we introduce λ as a streamwise lengthscale. With this lengthscale the Navier Stokes equations are transformed so that the streamwise momentum equation is λstu t + uu + vu = p + λre (u + λ u ). We assume a long wavelength approimations [, 3] u = U (,t) + ǫa(,t)u (,t) + O(ǫ ), p = P (,t) + ǫ P(,t) + ǫ ρa U d + O(ǫ3 ), where A(, t) is as et an unknown function that indicates the displacement of the centre streamlines. Substituting these epansions into the flow equations and imposing kinematic boundar conditions of zero normal velocit at the walls, we get a linearised KdV equation for the streamline displacement A given b [3] (γa) t βa = εγ (FF + FA + F A), (4) with γ(t) = U (,t) being the lower wall shear and β(t) = U (,t)d. The unperturbed flow upstream is taken to be unstead Poiseuille-like flow given b u = U (,t), v = and p = P (,t) = λre p (t) satisfing ReSt U t = p (t) + U with the boundar conditions U (±,t) = and the flu condition + U (,t)d = q(t). Here p is an unknown unstead pressure gradient and is determined to satisf the flu condition. This equation is solved using a Chebshev collocation method with Clenshaw-Curtis quadrature used to approimate the flu conditions (see for eample [5]). With reference to figure we solve (4) with the boundar function F() =.5 {tanh[α( 6)] tanh[α( 9)]}, 5

6 which approaches the desired sudden epansion as α. We use eponential time stepping with periodic Fourier spectral discretisation for spatial derivatives as described in [6]. We solve the equation for in the interval [,8] which is longer than the channel used in the numerical simulations but reduces the effect of the downstream boundar. (d) (c) (b) (a) Figure 6: Displacement of centre streamlines A(, t) given b the solution of (4). The plot of A is given at t = (a), t = (b), t = 3 (c) and t = 4 (d). Figure 6 shows the streamline displacement A as a function of at the different times and reveals that a wave develops at the epansion and is propagated downstream. There is another smaller wave that develops at the point where the channel width returns to. The speed at which these waves propagate is of order and small compared to KH-waves which have speeds of order St calculated from solutions of the Navier Stokes equations. The waves computed from the KdV equation have wavelengths that are in the regime of the vorte waves that develops before the flow begins to shed KH-waves. 5. Linear Stabilit Theor In the long channel flows we are considering, there are significant regions of almost parallel flow so it is natural to consider whether the genesis of the KH-wave can be eplained b considering stabilit of a parallel flow. There is a substantial classical theor that considers linear disturbances to a parallel flow although its application to smmetric channel flow has in recent times been deprecated because predicted growth rates when the underling flow is unstable are so small that the time or travel distance needed for a disturbance to develop are unrealisticall long. The model we eamine here takes the unperturbed flow to be u () = 3 ( σ)( ),, (5) with wall shear at = given b u () = 3(σ ) so that the profile can represent reverse flow near one wall when σ >. The choice of profile is significant because it preserves the flu through a channel as σ is varied. Eample profiles are shown in figure 7. When the parallel flow, (5), is perturbed as u u () + e ik( ct) ψ (), v ike ik( ct) ψ(), (6) the function ψ() of the perturbation satisfies an Orr-Sommerfeld equation (u o cst)d ψ u ψ = ikre D4 ψ, (7) 6

7 u (σ =) u (σ =.) u (σ =.5) u (σ =.) Figure 7: Eample parallel flow velocit profiles for σ =,.,.5,. where D = d /d k and ψ(±) = ψ (±) =. It is well known that the smmetric case, σ =, becomes unstable to small disturbances (c i = Im(c) > ) at a critical value Re 3848 (because of scaling in (5), value is /3 of 577) but also that at Renolds numbers above this critical value, the amplification of disturbances is minute, for eample at Re = 4, the maimum amplification per ccle of the disturbance is.56, impling near 48 ccles for a disturbance to grow b one order of magnitude. However, it is not generall appreciated that the smmetric case is in a sense an anomalous case, and that when the parallel flow is asmmetric, the predictions of linear stabilit theor are much different. In figure 8 the critical renolds number for loss of stabilit is shown as the asmmetr parameter σ is varied. For small values of σ the critical Renolds number increases but for values where reverse flow eists at one wall, the critical Renolds number decreases to less than. Re σ Figure 8: Critical Renolds number for instabilit as the asmmetr parameter σ is varied. In order to compare the predictions to computed Navier Stokes flows it is useful to observe that the disturbance is essentiall like e kc it e ik( crt), so that the wavelength of a disturbance is λ = π/k, the frequenc of a disturbance is Ω = kc r /πst and the amplification per ccle, denoted ζ, is ζ = ep(πc i /c r ). At Re = 6 and σ =.5, the maimum growth per ccle is 3 and an order of magnitude growth occurs in onl a few ccles. The growth rate is even larger for higher values of σ, see figure 9 so that near σ =, an order of magnitude growth occurs in onl one ccle. Furthermore, the frequenc of disturbances is ver high, for eample in a periodic flow with Re = 6, St =.5, σ =, Orr- Sommerfeld solutions predict the frequenc of a disturbance to be around 9 times the imposed oscillator frequenc and a non-dimensional wavelength near Discussion The KH-wave illustrated above is, we believe, a previousl unreported flow pattern associated with unstead flow through channels. Our work so far tends to eclude modelling the wave as the result of a long wave-length disturbance to a rotational core flow. Our preliminar results when modelling the wave using linear stabilit theor show some correspondence but are not conclusive. Computations of the Navier Stokes equations at Re = 6, St =.5 7

8 St c i k.5. St c r ζ σ (i).5.5 σ (ii) Figure 9: (i) Contours of Stc i as σ and k are varied. The flow is unstable where c i >. The dashed line shows values of wavenumber k where c i is maimum for fied σ. (ii) Real and Imaginar parts of maimum eigenvalue Stc and corresponding growth per ccle, ζ, for Re = 6 as σ is varied. indicate the KH-wave has approimate wavelength 3-5 and frequenc around the imposed frequenc whereas the Orr-Sommerfeld prediction for σ = is wavelength 6 and frequenc multiple 9. Of course the Navier Stokes solutions are for an evolving flow so perhaps this degree of correspondence is the best that will emerge. In addition, studing the distribution of vorticit for the Navier Stokes solutions suggest that the emergence of this fast travelling wave also follows a Kelvin-Helmholz roll instabilit of the vorticit laer between vortices near a wall and the main flow but there are still questions as to wh such an instabilit occurs near specific vortices and not near all. Acknowledgement To our knowledge, the first computation of a KH-wave was in unpublished work b IJS and T.-H. Hwu. DK is grateful for the award of a Rhodes Scholarship References [] I. J. Sobe, Observation of waves during oscillator channel flow Journal of Fluid Mechanics, 5, (985). [] O. R. Tutt and T. J. Pedle, Oscillator flow in a stepped channel, Journal of Fluid Mechanics, 47, 97 4 (993). [3] O. R. Tutt and T. J. Pedle, Unstead flow in a nonuniform channel: A model for wave generation, Phsics of Fluids, 6, 99 8 (994). [4] T.-H. Hwu and I. J. Sobe and B. J. Bellhouse, Observation of concentration dispersion in unstead defelected flows, Chemical Engineering Science, 5, (996). [5] L. N. Trefethen, Spectral Methods in MATLAB, SIAM, (). [6] A.-K. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff pdes, SIAM Journal of Scientific Computing, 48, 4 33 (3). 8