Pattern formation, vortex instabilities and dynamics in lid-driven …ltces.dem.ist.utl.pt/lxlaser/lxlaser2012/upload/393... · · 2012-06-20Pattern formation, vortex instabilities

16th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 09-12 July, 2012

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Pattern formation, vortex instabilities and dynamics in lid-driven cavities

Tanja Siegmann-Hegerfeld1,*, Stefan Albensoeder2, Hendrik C. Kuhlmann3

1: Institute of Chemical Engineering, Vienna University of Technology, Austria

2: Voith Engineering Services GmbH Aerospace, Germany 3: Institute of Fluid Mechanics and Heat Transfer, Vienna University of Technology, Austria

* Correspondent author: [email protected] Abstract Flow structures in lid-driven cavities are investigated numerically and experimentally. Depending on the geometry and the Reynolds numbers a variety of vortex instabilities originate in this classical system. For the one-sided driving and large cross-sectional aspect ratios (Γ > 1.207) a stationary mode, called C4

e, has been predicted numerically and observed experimentally (Γ = 1.6, spanwise aspect ratio Λ ≈ 11) by former investigations. Experimentally the transition occurs gradually from the two-dimensional basic flow to the three-dimensional flow pattern. The transition starts at a considerably lower Reynolds number than predicted by the linear stability analysis for the spanwise infinite cavity. Streak-line photographs and PIV vector maps are presented to illustrate the structure of the flow pattern of the mode. LDV measurements confirm an imperfect bifurcation for the investigated geometry and clarify the deviations between experimental and numerical critical Reynolds numbers. Three-dimensional simulations conducted for the finite-span geometry show a good agreement with corresponding measurements. Therefore, the numerical predicted C4

e-mode can be considered experimentally confirmed. The numerical and experimental data lead to the conjecture that the premature onset of the instability is caused by strong secondary flows induced by the cavity end walls. 1. Introduction Lid-driven cavity flows offer various phenomena which are of fundamental interest for fluid dynamics. The simple geometry and boundary conditions of this classical system enable well-controlled and accurate investigations into vortex flows. Owing to the simple geometry, a large body of numerical work has been published on the lid-driven cavity problem. However, experimental results are relatively sparse. The purpose of this work is to partially close this gap. For an overview on lid-driven cavity flows we refer to Shankar and Deshpande (2000). Depending on the geometry and on the Reynolds numbers a variety of different flow patterns can be generated in lid-driven cavities. Here, we focus on the transition from essentially two- to three-dimensional flow structures due to symmetry-breaking bifurcations. The linear stability problem for the two-dimensional basic flow is governed by a three-dimensional parameter space, given by the two Reynolds numbers and the aspect ratio, and it has been considered by Albensoeder and Kuhlmann (2003) and Albensoeder (2004). In the case when only a single lid is moving (one-sided driving) they predicted four different three-dimensional instabilities which arise due to centrifugal mechanisms; two oscillatory and two stationary instabilites (Albensoeder et al. 2001). Numerical investigations of Theofilis et al. (2004), likewise, result in four modes for the one-sided driving, but three modes are predicted to be oscillatory while only one is stationary. The critical parameters agree only for one of the stationary modes and only for Γ ≈ 1. Experiments of Albensoeder et al. (2001), Blohm and Kuhlmann (2002), Siegmann-Hegerfeld et al. (2008a/b) and Siegmann-Hegerfeld (2010) validate the modes as predicted by Albensoeder et al. (2001) and Albensoeder (2004). For larger aspect ratios (Γ > 1.207) the second stationary mode, called C4

e, was observed in agreement with Albensoeder’s (2004) predictions. In all cases the characteristics of the flow structures, the critical wave numbers and the oscillation frequencies show a good agreement with the numerical predictions. However, for two modes larger deviations in the critical Reynolds numbers are found. The reasons for these deviations can be various, but experimental imperfections


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Fig. 1 Sketch of the geometry of the cavity, coordinate system, and definition of the rotation of the side walls. and deviating boundary conditions have to be considerd, in particular. For the stationary C4

e-mode the critical Reynolds number deviates by ~30%. The present investigation intends to clarify the cause of this deviation. Photographs and PIV vector maps are presented to describe and measure the flow of the C4

e-mode. Finally, we provide results of full numerical simulations for periodic and for no-slip boundary conditions and compare these to the results of LDV measurements. 2. Experimental setup and numerical methods Only basic information on the experimental setup is given in this section. For a detailed description we refer to Siegmann-Hegerfeld (2008a). The flow in the cavity is driven by two large cylinders rotating about their horizontal axes. The cylinders are located parallel to each other in a distance d (width) and form two opposing walls of the cavity. Due to the shape of the cylinders, the moving walls are slightly curved. Fig. 1 shows a sketch of the cavity, where the moving walls are located on the left and on the right. The coordinate system is located in the center of the cavity. The anti-clockwise rotation of the cylinders is defined as positive. A top cover, a bottom wall and two end walls define the cavity. To allow visual access the bottom wall and the end walls are made of Perspex. All parts are mounted inside an outer container filled with silicone oil (kinematic viscosity ν = 20 mm²/s at 25°C). For flow visualization non-isotropically-shaped aluminium flakes are illuminated by a halogen light sheed. Photograhs are taken with a digital reflex camera (Nikon D70) and PIV vector maps are obtained by using a standard 2d-PIV System (Dantec Dynamics, ND:YAG Laser 200mJ/15Hz each cavity, CCD camera 2Mpix). For PIV and LDV white glass spheres (Potter Spherical 110P8CP00) are used as seeding. The dimensionless parameters of the cavity are the two Reynolds numbers related to the moving walls (Re), the cross-sectional aspect ratio Γ (width d to heigth h) and the span aspect ratio Λ (length l to heigth h). The two Reynolds numbers are defined as

νdu

Re 2,12,1 = , (1)

where u1,2 are the velocities of the corresponding walls (indicated by the subscripts), d is the arithmetic mean of the two extremal cavity widths in x-direction and ν is the kinematic viscosity of the liquid. The length scale h, the viscous velocity scale ν/h and the viscous time scale h²/ν are used to present the results in dimensionless units. Reynolds numbers without subscripts are used to

Re1 h dmin

l

Γ/2 0

x

Re2

Λ/2

dmax

y

z


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(a)

(b)

(c)

Fig. 2 Streak lines of different three-dimensional flow patterns for Γ = 1.6 in the (x,z)-plane y ≈ 0. The Reynolds numbers are Ree = 600 (a), Rep = 500 (b) and Rea = 300 (c). C4

e-mode, the top wall moves into the plane (a). Cp-mode, both walls move into the plane (b). E2

a-mode, the top wall moves out of the plane and the bottom wall moves into the plane. indicate either one-sided driving Ree: = -Re1 or Re1, or equal wall velocities with Rep : = - Re1 = Re2 (symmetric parallel driving) or Rea : = Re1 = Re2 (symmetric antiparallel driving). The experimental data are compared with numerical results for rectangular cavities in which the curvature of the moving walls is neglected. Linear stability data were obtained by a finite-volume method using periodic boundary conditions in spanwise direction (Albensoeder and Kuhlmann 2002a, b and 2003). For time-dependent three-dimensional simulations a code based on Chebyshev collocation has been adopted (Albensoeder and Kuhlmann 2005). Detailed information on the numerical methods can be found in Albensoeder (2004). 3. Results The C4

e-mode was first mentioned by Kuhlmann et al. (1997) and, thereafter, investigated in more detail by Albensoeder et al. (2001). According to Albensoeder et al. (2001) the C4

e-mode arises for one-sided driving when the aspect ratio is large (Γ > 1.207). The stationary C4

e-mode is driven by centrifugal mechanisms. The three-dimensional finite-amplitude perturbation flow deflects the basic vortex periodically up and down in positive and negative y-direction along z with the critical wave lenghts λc, such that the total flow appears as a stationary wavy vortex. For the experimental investigations an aspect ratio of Γ = 1.6 is used. Streak lines of the fully developed C4

e-mode are shown in Figure 2a. For two-sided driving and depending on the Reynolds numbers and the aspect ratio two other supercritical flow patterns may arise for Γ = 1.6 due to different instability mechanism. Examples are presented in Fig. 2b and 2c. Fig. 2b displays the structure of the Cp-mode emerging at larger aspect ratios and Fig. 2c shows the cellular structure of the E2

a-mode. The notation for these modes is the same as introduced by Albensoeder. Examples of the two-dimensional basic flow for Γ = 1.6 in the (x,y)-plane are presented in Figure 3. Fig. 3c shows the basic flow of the one-sided driving. The flow consists of a basic vortex adjacent to the moving wall and a secondary eddy. The C4

e-mode also occurs at the asymmetric parallel driving when increasing Re2 above the critical Reynolds number. An expample of the asymmetric parallel driven basic flow is shown in Fig. 3b. Only if the basic flow is nearly or exact symmetrical


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(a) (b) (c)

(d) (e) (f)

Fig. 3 Streak lines of the basic flow for Γ = 1.6 in the (x,y)-plane z ≈ 3.5. The Reynolds numbers are Rep = 400 (a), (Re1,Re2) = (-100,400) (b), Ree = 300 (c), (Re1,Re2) = (200,400) (d), (Re1,Re2)=(200,300) (e) and Ree = 200 (f). The walls on the right- and left-hand side move downward (a, b). The wall on the left-hand side moves downward (c). The left wall moves upward while the right wall moves downward (d-f). with respect to the midplane (x = 0) as shown in Fig 3a, the Cp-mode (Fig. 2b) appears as supercritical pattern. When the flow is asymmetric antiparallel driven (e.g. Fig. 3d and 3e, Re1 ≠ Re2) the strength of the asymmetry determines whether the C4

e-mode (Fig. 2a) or the E2a-mode (Fig.

2c) will be generated. Fig. 3f shows the basic flow of the symmetric antiparallel driving (Re1 = Re2). An increase of Re2 (Fig. 3e) leads to a change of the basic flow. The asymmetry of the flow structure increases until the vortex of the symmetrical state changes to a asymmetrical two-vortex flow, where even the antiparallel-downstream-secondary-eddy (Koseff and Street 1984) becomes visible (Fig. 3d). Based on this quasi-two-dimensional flow structure, the C4

e-mode appears when Re2 is increased above the critical Reynolds number. Consequentially, it is assumed that the C4

e-mode is generated if the two-vortex-flow (Kuhlmann et al., 1997) exists as the basic state. When the two vortices of the basic flow, on the other hand, merge due to an increase of the symmetry, the E2

a-mode arises at sufficiently large Reynolds numbers. Experimentally we observe the supercritical pattern of the C4

e-mode at Reynolds numbers considerably higher (~30%) than the critical Reynolds number predicted by a linear stability analysis (Albensoeder 2004). Experimentally, no clear critical point for the onset of the C4

e-mode is found. Rather, three-dimensional flow components are present in the spanwise center of the cavity even at Reynolds numbers which are distinctly smaller than the numerically predicted critial one (Re < Rec). Increasing the Reynolds number the three-dimensional flow structures become more dominant. In contrast to this, the stability boundary can be clearly detected for smaller aspect ratios (Γ < 1.2). At small aspect ratios (e.g. Γ = 1) we observe significant changes of the flow structure altering from the basic flow to the supercritical pattern, when the Reynolds numbers are increased in small steps (Δ Re = 5) above the critical Reynolds numbers. For that reason the critical Reynolds numbers can be detected with an accuracy of ΔRe = 5. In contrast to that the detection of the critical Reynolds numbers of the C4

e-mode is made difficult due to the gradual transition. In Figure 4 streak line photos of the flow structures in the (x,z)-plane are presented. The Reynolds number is increased from Ree = 300 (a) to Ree = 1100 (f). Fig. 4a shows the quasi-two-dimensional basic flow. Side wall effects are clearly visible and even propagate far into the bulk of the cavity.


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(a)

(b)

(c)

(d)

(e)

(f)

Fig. 4 Streak lines of of the basic flow (a), the transition to the three-dimensional flow pattern of the C4

e-mode (b-d), the fully developed C4

e-mode (d) and time dependent flow structures at high Reynolds numbers (e,f) for Γ = 1.6 in the (x,z)-plane y ≈ 0. The Reynolds numbers are Ree = 300 (a), Ree = 400 (b), Ree = 500 (c), Ree = 600 (d), Ree = 900 (e) and Ree = 1100 (f). The top wall moves into the plane. Only a relatively small area in the spanwise center (-1.4 < z < 1.4) seems to be free of three-dimensional effects. The numerically calculated critical Reynolds number is Rec = 479.97 for Γ = 1.6. Changes of the flow structure in the center of the cavity become apparent even at Ree = 400 (Fig. 4b), due to reflexions of the visualization particles aligning in the flow. The deviation from the basic flow can be recognized by two white oval areas, near the driven wall in the center of the cavity. The structure becomes apparent at Ree ≈ 350 and is more and more pronounced with increasing Reynolds numbers (Fig. 4c). The three-dimensional pattern becomes fully developed between Ree = 500 and Ree = 600. Indicative of it are two vertical lines giving the impression that the flow would consist of three cells. The wave number of kexp ≈ 1.73 agrees well with the numerically computed critical wave number kc

num = 1.7426. The main vortex is periodically modulated in the form of an alternating expansion and contraction in z-direction. This observation agrees well with the numerical prediction. The flow is stationary up to Ree ≤ 800. At higher Reynolds numbers (Fig. 4e) the flow becomes time-dependent, although the flow pattern remains visible. When the Reynolds number is further increased above Ree ≈ 1050, a different flow structure arises and the supercritical pattern of the C4

e-mode is destroyed in favor of smaller scale structures with a preferred spanwise wave number (Fig. 4f) which appear to have a chaotic dynamics.


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(a)

(b) (c) (d)

(e) (f)

Fig. 5 Streak lines of the C4

e-mode for Γ = 1.6 and Ree = 600 in the (x,z)-plane y ≈ 0 (a) and in the (x,y)-plane (b-f). The top wall moves into the plane (a). The wall on the left-hand side moves downward (b-f). The z-positions of (b-f) are marked in (a, bottom side). Additionally the z-positions of Fig. 6 are marked at the top side. The spanwise alternating deformation of the main vortex and the periodic shift of the vortex center of the C4

e-mode are shown in Figure 5. Fig 5a displays the supercritical pattern in the (x,z)-plane. In 5b-5f the structures in the (x,y)-plane are presented at different z-positions. The z-positions are indicated at the lower side of the photo in Fig. 5a. The strongest contraction of the main vortex is visible in Fig. 5b. The secondary eddy of the basic flow has disappeared. In Fig. 5c the main vortex is extended again and the flow structure is similar to the basic flow. The neighboring secondary eddy is clearly visible again. In Fig. 5d the cross-section area of the main vortex takes its maximum and the area occupied by the secondary eddy is minimized. Further in z-direction the process is reversed, leading to the basic flow like structure in Fig. 5e and finally to the strongest contraction of the main vortex in Fig. 5 f. In Figure 6 corresponding PIV vector maps of the flow in the (x,y)-plane are presented. Starting at one end of the cavity at z ≈ -5.4 (Fig. 6a) the position of one to the next vector map (Fig. 6a, 6c-6h) is shifted about a quarter wave length in positive z-direction. The z-positions of 6a-6h are marked in Fig. 5. The vector maps 6f, 6g and 6h are corresponding to the photos 5b, 5c and 5d. As the z-positions of 6b-6f and 5b-5f are shifted about one wave length λ in z-direction, the flow structures in Fig. 6b-6f should coincide with the photos 5b-5f. A comparison of the structures shows

6a 6d 6c 6e 6b 6f 6g 6h

b f c d e


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(a) (b) (c)

(d) (e) (f)

(g) (h)

Fig. 6 PIV vector maps of the C4

e-mode for Γ = 1.6 and Ree = 600 in the (x,y)-plane at different z-positions. The z-positions are marked in Fig. 5. The wall on the left-hand side moves downward. The resolution of the vector maps decreases from (a) to (h), which might result in a slight change of impression despite similar flow patterns. The vector maps are statstics resulting from 100 double pictures (reference vector v = 50mm/s). a good qualitative agreement. In the experiments only the z-position of the light sheet is changed, not the position of the camera. Therefore, the resolution of the PIV vector maps decreases from 6(a) to 6(h). A slightly changed impression can originate from the decreasing resolution, despite looking at the same flow structures. To shed some light on the experimentally observed imperfect bifurcation of the primary instability we carried out numerical computations for a reference point and for another position for both spanwise periodic boundary conditions and for rigid end wall conditions corresponding to the experimentally used span aspect ratio Λ = 10.85. Initially, a reasonable reference point for the measurements was determined numerically (x,y,z) = (x*,y*,z*) for periodic boundary conditions. To guarantee a significant three-dimensional modulation of the flow, the position (x*,y*,z*) = (-0.562,-0.226,z*) was selected where the velocity w takes its global maximum. To clarify the influence of the end walls, velocity profiles have been computed as a function of z for different Reynolds numbers. Results for w = w(x*,y*,z) (a) and w = w(0,0,z) (b) are shown in Figure 7. Results for periodic boundary conditions are indicated by dashed lines and results for Λ = 10.85 are marked by continuous lines. The computation was carried out for the numerically calculated critical wavelength λ = 3.595. The colors mark different Reynolds numbers: Ree = 300 (brown), Ree = 400 (green), Ree = 500 (red), Ree = 600 (blue), Ree = 700 (black). At first, the profiles of the reference position in Fig. 7a are addressed. The spanwise velocity component w at Ree = 300 is almost zero for -4 < z < 4 which indicates that the flow is still quasi-two-dimensional, larger velocities appear close to the end walls of the cavity (-4 > z > 4) indicating strong secondary flows. At Ree = 400 a


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(a) (b)

Fig. 7 Computed profiles of the velocity component w for Γ = 1.6 and λ = 3.595. The Reynolds numbers are Ree = 300 (brown), Ree = 400 (green), Ree = 500 (red), Ree = 600 (blue) and Ree = 700 (black). (a) w(x*,y*,z) and (b) w(0,0,z). Periodic boundary conditions are indicated by dashed lines and simulations for Λ = 10.85 are denoted by continuous lines. slight deflection is already developed and at Ree = 500 a higher amplitude is visible. The critical Reynolds number due to the linear stability analysis is Rec = 479.97 for Γ = 1.6. The profiles for Ree = 500 for periodic boundary conditions and for the finite cavity show a substantial difference in their characteristics as well as a large deviation in the amplitudes. The perturbation amplitude in the center is smaller for no-slip end-wall conditions than for periodic boundary conditions. At the end walls of the cavity the amplitude of w is higher for no-slip end wall conditions than for the periodic boundary conditions. This characteristic is, therefore, attributed to a strong end-wall effect. With higher Reynolds numbers the profiles converge increasingly. Only slight deviations remain at Ree = 700. The computations for the finite cavity agree very well with the visual observations. Increasing Ree up to 500 the three-dimensional flow structure develop gradually and appear fully developed at Ree = 600. The spanwise velocity component w of the profiles shown in Fig 7b exhibits a similar behavior. Indeed, the deviations close to the end walls are larger. Apart from that, the profiles exhibit more complexity than along the reference ray. Measured spanwise velocities are included by symbols in Figure 8. The profiles are presented in Fig. 8a for Ree = 300 and Ree = 400, in Fig. 8b for Ree = 500, in Fig. 8c for Ree = 600 and in Fig. 8d for Ree = 700. The numerical computations were carried out for positive Re1 and the measurements for negative Re1. Accordingly, the measuring point was adapted by using the symmetries of the system (x*,y* = -0.562,0.226). The measured velocities show a good agreement with the computed data. Even though the profiles deviate slightly in amplitude, the experimental data coincide, in particular for Ree = 500, very well with the computations for the rigid end wall conditions. Experimental data along the other ray (0,0,z) are presented in Figure 9. The Reynolds numbers are Ree = 500 (Fig. 9a) und Ree = 700 (Fig. 9b). Again, the experimental data are in good agreement with the numerical data and, as for the reference point, the measured profile for Ree = 500 (Fig. 9a) agrees very well with the simulations for the finite cavity. Based on the comparision of measured and computed data the three-dimensional flow can doubtlessly be identified as the numerically predicted C4

e-mode. Further LDV measurements were carried out to confirm the imperfect bifurcation of the C4

e-mode quantitatively. On account of the differing boundary conditions due to the end walls of the container


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(a) (b)

(c) (d)

Fig. 8 Measured (symbols) and computed profiles (lines, λ = 3.595) of the velocity componente w for Γ = 1.6 at w(x*,y*,z). Periodic boundary conditions are indicated by dashed lines and simulations for Λ = 10.85 are denoted by continuous lines. (a) Ree = 300 (open lozenge) and Ree = 400 (blacklozenge), (b) Ree = 500, (c) Ree = 600 and (d) Ree = 700. in the experimental set up, the position z* has to be determined relative to the pattern. The position closest to the center (z = 0), given by z = -1.25, was selected for the measurements. The bifurcation diagram of the C4

e-mode is presented in Figure 10. The open symbols (lozenge) are the data of the first measurement. Data of a repetition experiment are included by black symbols (black lozenge). The vertical line indicates the linear-stability boundary. The continuous line marks simulations for the reference point for periodic boundary conditions. In addition, the large symbols indicate the maximum velocities of the profiles in Fig. 8 (+ measured and ○ simulations for Λ = 10.85). The measured data show a good agreement with the simulations for the finite cavity. The imperfect bifurcation becomes evident at Ree ≈ 270 by a velocity w slightly increasing with respect to its value for pure two-dimensional flow. At Ree ≈ 350 the slope of w(Re) becomes larger and increases even more at Ree ≈ 525. Close to Ree = 550 the curvature of w(Re) changes sign. It is in this range (Ree = 500 to Ree = 600) where the supercritical pattern and the velocity profiles develop markedly. The experimental and numerical evidience leads to the conclusion that the secondary flows induced by the presence of the end walls cause a premature onset of the instability. Further to this, the relatively long cavity with span aspect ratio Λ = 10.85 is still to short for a clear detection of the critical Reynolds numbers of the long-wave C4

e-mode.


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(a) (b)

Fig. 9 Measured (symbols) and computed profiles (lines, λ = 3.595) of the velocity componente w(0,0,z) for Γ = 1.6. Periodic boundary conditions are indicated by dashed lines and simulations for Λ = 10.85 are denoted by continuous lines. (a) Ree = 500 and (b) Ree = 700.

Fig. 10 Measured (black and open lozenge; open lozenges indicate data of an independent validation measurement) and computed bifurcation of the C4

e-mode for Γ = 1.6 (w(x*,y*,z*) = maxz w(x*,y*,z)). The vertical line indicates the linear-stability boundary. The preset numerical wave lenghts were λ = 3.595. The continuous line indicates simulations for periodic boundary conditions. The big symbols indicate the maximum velocities of the profiles in Fig 8 (+ measured and ○ simulations for Λ = 10.85). 3. Summary and discussion The stationary C4

e-mode is characterized by an expansion and contraction of the main vortex and a displacement of the vortex center along its axis. This characteristic is similar to the sturcture of the Cp-mode. The C4

e-mode occurs as primary instability at larger aspect ratios (Γ > 1.2) when the basic flow is driven one-sided or asymmetric parallel. The C4

e-mode also occurs for asymmetric antiparallel driving depending on the strenghts of the asymmetry. It is conjectured that the C4

e-mode is generated when the two-vortex flow exists as the basic flow. For the aspect ratio Γ = 1.6 and a span aspect ratio Λ ≈ 11 the C4

e-mode shows a gradual transition from the two-dimensional basic


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flow to the three-dimensional flow pattern. For that reason, the detection of the critical Reynolds number by visual investigations was difficult. The three-dimensional flow structure emerges at Re ≈ 350 while a linear stability analysis for periodic boundary conditions predicts a critical Reynolds number of Rec = 480 (Γ = 1.6). The imperfect bifurcation was confirmed by LDA measurements. In fact at Re ≈ 270 the velocity w slightly increases with respect to its value for pure two-dimensional flow. On account of the numerical and experimental data it is concluded that secondary flows cause the premature onset of the instability. The results show that the large span aspect ratio of Λ ≈ 11 is still too small for a precise experimental detection of the critical Reynolds numbers. By comparison of measured and computed velocity profiles for no slip and for periodic boundary conditions we could unambiguously identify the experimentally observed three-dimensional structure as the C4

e-mode predicted by Albensoeder (2004). References Albensoeder S, Kuhlmann HC, Rath HJ (2001) Three-dimensional centrifugal-flow instabilities in the lid-

driven cavity problem. Phys. Fluids 13:121-135 Albensoeder S, Kuhlmann HC (2002a) Linear stability of rectangular cavity flows driven by anti-parallel

wall motion of two facing walls. J Fluid Mech 458:153-180 Albensoeder S, Kuhlmann HC (2002b) Three-dimensional instability of two counter-rotating vortices in a

rectangular cavity driven by parallel wall motion. Eur J Mech B/Fluids 21:307-316 Albensoeder S, Kuhlmann HC (2003) Stability balloon for the double-lid-driven cavity flow. Phys. Fluids

15:2453-2456 Albensoeder S (2004) Lineare und nichtlineare Stabilität inkompressibler Strömungen im zweiseitig

angetriebenen Rechteckbehälter. Cuvillier Verlag, Göttingen Albensoeder S, Kuhlmann HC (2005) Accurate three-dimensional lid-driven cavity flow. J Comput Phys

206:536-558 Blohm C, Kuhlmann HC (2002) The two-sided lid-driven cavity: Experiments on stationary and time-

dependent flows. J Fluid Mech 450:67-95 Koseff JR, Street RL (1984) The lid-driven cavity flow: A synthesis of qualitative and quantitative

observations. J Fluids Eng 106:390-398 Kuhlmann HC, Wanschura M, Rath HJ (1997) Flow in two-sided lid-driven cavities: Non-uniqueness,

instabilities, and cellular stuctures. J Fluid Mech 336:267-299 Shankar PN, Deshpande MD (2000) Fluid mechanics in the driven cavity. Annu Rev Fluid Mech 32:93-136 Siegmann-Hegerfeld T, Albensoeder S, Kuhlmann HC (2008a) Two- and three-dimensional flows in nearly

rectangular cavities driven by collinear motion of two facing walls. Exp Fluids:45:781-796 Siegmann-Hegerfeld T, Albensoeder S, Kuhlmann HC (2008b) Experiments on the stability, structure, and

dynamics of flows in one- and two-sided lid-driven cavities. 14th Int Symp on Applications of Laser Techniques to Fluid Mechanics 1277

Siegmann-Hegerfeld T (2010) Wirbelinstabilitäten und Musterbildung in geschlossenen Rechteckbehältern mit tangential bewegten Wänden. Ph.D Thesis, Vienna University of Technology, Austria

Theofilis V, Duck PW, Owen J (2004) Viscous linear stability analysis of rectangular duct and cavity flows. J Fluid Mech 505:249-286

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Pattern formation, vortex instabilities and dynamics in lid-driven …ltces.dem.ist.utl.pt/lxlaser/lxlaser2012/upload/393... · · 2012-06-20Pattern formation, vortex instabilities