4.2 Binary Fluid Convection
4.2.2 Travelling Wave Convection
For the investigation of travelling wave convection, a 25%wtmixture of ethanol-water was chosen with a separation ratio ofψ=−0.087 [45, 63] at T= 300 K.
The Prandtl number isσ≃18 and the Lewis number is found to beL≃0.0061.
The experimental procedure was to drive the system into the state of TW at r > rco initiating a fully developed travelling wave state and lowering the Rayleigh numberr in small steps thereafter until the transition to conduction at rsn was reached. Subsequently, the critical Rayleigh number, rco, for the
1.05 1.1 1.15 1.2 0.2
0.3 0.4 0.5 0.6 0.7
r
λ
λ1
λ2+0.3 λ3+0.2
Figure 4.12: The eigenvalues λi of the first three POD mode versusr of TW convection in a 25%wt ethanol-water mixture withψ=−0.087.
onset of TW convection was determined. In approachingrco from below, the fluid was allowed to relax approximately 50τv while the relaxation time in between travelling wave states was 30τv with the vertical thermal relaxation timeτv≃83 s.
This experiment has to be regarded as a borderline case for the application of µPIV because the moving roll structure does not allow for an effective av-eraging. Note that the ensemble correlation evaluation could be realised in a co-moving frame but this would narrow the resulting velocity field propor-tional to the phase velocity of the TW. Therefore, the data is evaluated by means of a two-mode reconstruction based on the POD (Fig. 4.13). However, the noise level is found to be substantially larger compared to stationary con-vection due to the separation of the basal concon-vection structure into two POD modes. The POD evaluation requires “dense” tracer images. In the present case approximately half the measurement data had to be discarded which is the explanation for the comparatively few data points of the results.
Figure 4.12 presents the eigenvalues of the first three POD modes. Mode one and two cover the structure information whereas the third represents the most energy rich mode of the background. Modes three and four are comparable to modes two and three of the 40%wtmixture and might be regarded as the non-random measurement noise according to the argumentation of the preceding section. The energy content of the first mode varies between 60 and 70%
while the second one contains up to 12% of the total energy, depending on the frequencyω of the TW. The first and second mode are of similar morphology, shifted by half a wavelength against each other. In the limit of large enough
4.2 Binary Fluid Convection
0 0.5 1 1.5 2
−0.5 0 0.5
x/d
z/d
0 0.5 1 1.5 2
−0.5 0 0.5
x/d
z/d
Figure 4.13: The velocity field representation of the first and the second POD eigenmode of the flow atr= 1.08(2).
image time series the eigenvalues of both should become equal, forming a mode pair. Otherwise, the increasing TW frequency results in an increasing modulation of the flow structure, due to the limited recording time. Therefore, the energy content of the second mode rises with increasingω. The noise level, represented by the third and higher order modes, remains constant as long the TW state persists.
Increasing r, the conductive state loses stability atrco= 1.13(5) (dotted line in Fig. 4.14), which agrees with Eq. 2.13, yielding 1.094 within 4%. Subse-quently, loweringrleads to a TW state at decreasing amplitude and increasing frequency. The transition to conduction is found at rsn= 1.03(4) via a state of confined TW which will be addressed separately below. Figure 4.14 depicts the first two modes of the lateral Fourier decomposition of the TW state for both velocity components. The vertical component w is taken at mid-height and u is extracted at its maximum amplitude position which is the same as for the stationary case (z/d=±0.28). Note that the inversion symmetry of the w-profile is broken, leading ton= 2 being the second mode inw. |u1|and|w1| are found to be closely below the pure fluid reference in agreement with two-dimensional numerical simulations [9]. The slightly larger deviation of|w1|has to be attributed to the wave number which, within experimental uncertainty, is constant for the Rayleigh numbers measured, k= 2.5(2). The small value
1 1.05 1.1 1.15 1.2 0
1 2 3 4 5
r un[κ/d]
5⋅|u2|
|u1|
→
←
1 1.05 1.1 1.15 1.2
0 1 2 3 4 5
r wn[κ/d]
5⋅|w
2|
|w1|
→
←
Figure 4.14: The first and second Fourier mode amplitude of u and w in the TW state of the 25%wt ethanol-water mixture with ψ=−0.087, σ≃18, and L≃0.0061; the error bars denote the uncertainty of the approximation.
of the wave number is unexpected as the TW state characteristics are known to closely accord the pure fluid reference suggesting kpc= 3.005 [99]. However, the source of this deviation, similar to that of the stationary Rayleigh mode, has not been resolved.
The second mode exhibits considerable scatter in both velocity components as a result of the increased noise level of the measurements.
The frequency of the TW is determined directly from the zero-crossing of
4.2 Binary Fluid Convection
1 1.05 1.1 1.15
1 1.05 1.1 1.15 1.2 1
r
ω
Figure 4.15: The frequency of the TWω versus r.
the lateral velocity profiles at mid-height. Normalised with the predicted Hopf frequencyωH= 6.349 according to Eq. 2.14, the measured frequency of the TW at onset rco is approximately 0.05(1)ωH which increases to about 0.34(2)ωH at r= 1.03(5). Both values nicely agree with the results of numerical simula-tions [9].
In summary, the current measurements do not indicate considerable differ-ences from the theoretical predictions for the stationary Rayleigh mode. The underlying explanation is that the concentration field is confined to the very thin boundary layers and the efficiency of mixing in the bulk of the Rayleigh mode masks the binary nature of the fluid. These findings, however, confirm the results of numerical simulations [9].