Transient Dynamics in Counter-Rotating Stratified Taylor-Couette Flow

1. Introduction

The transition from laminar to turbulent fluid flow has been of interest for over a century. There are several states a fluid flow can exhibit beyond laminar before it becomes turbulent. These states have played a role in making significant contributions in industries such as automobile, civil, aerospace, and others. To address the instabilities arising from flow disturbances, a multitude of mathematical techniques have been developed within hydrodynamic stability theory [1,2].

Numerical and analytical methods, including bifurcation theory and other suitable approaches, have been employed to identify these flow patterns [3,4]. Among the various techniques, the transient growth method has gained extensive utilization in the exploration of subcritical transition in fluid flow, proving highly successful over several decades. This method has been invaluable in capturing and elucidating phenomena associated with stability problems in fluid flow that cannot be effectively addressed by linear stability theory alone [5,6,7,8,9].

The plane Couette flow and pipe Poiseuille flow are known to be stable for all Reynolds number within the theoretical mathematical framework of linear stability, but in practice these flows become turbulent for sufficiently large Reynolds number as accounted by Romanov[10], Davey[11], Drazin and Reid [12] and recent studies by others [13,14]. In practice the plane Poiseuille flow does exhibit an instability at a critical Reynolds number Rec ≈ 2300, but theoretically this phenomena is observed at a much larger Rec ≈ 5772 at which the so-called Tollmien-Schlichting wave depicting unstable mode appears[15,16]. The inconsistency in results has been attributed to the non-normality of the linearized Navier-Stokes operator. It is known that the non-normality is responsible for the spike (i.e. transient growth) in the growth rate within the linear regime that eventually trig- gers the transition of flow to turbulence. This behaviour is widely known as a subcritical transition or bypass transition[5,17,18]. The non-normal route to turbulence has raised many concerns in the literature by different schools of thought [7,19,20]. However, Reddy and Henningson[21] has objectively addressed the issues with insightful results for two and three dimensional plane Couette and Poiseuille flows. The utilization of the transient growth method for investigating subcritical bifurcations has witnessed widespread adop- tion and significant advancements over the last two decades with both theoretical and experimental justification[8,22,23,24].

The circular Couette flow often known as Taylor-Couette flow (TCF) has widely been studied experimentally and theoretically [25,26,27,28,29]. The TCF problem has been known to be one of the pioneering problems of hydrodynamic stability for several decades, since the early experimental and theoretical works of Taylor [30].

In 1965 Coles[31] experimentally investigated the transition of viscous flow in con- centric cylinders and reported that at certain Reynolds numbers where the linear theory predicted laminar flow, there exist spiral turbulence in the counter rotating flow [31]. In the following year, Van Atta [32] carried out similar experiments on the spiral transition of the flow and obtain related results. The result was also confirmed experimentally later in 1989 by Hegseth et al. [33]. In 2002 Prigent et al. [34] provided experimental results on spiral turbulence with agreement with Coles’[31] report.

Non-normality problem of the TCF linear operator was initially investigated by Greb- hardt and Grossmann in 1992 [26]. However, Hristova et al.[35] conducted a comprehensive investigation of this phenomenon using pseudospectral analysis. In 2002, Meseguer[36] examined the transient energy growth associated with the parametric regime reported by Coles[31] and observed significant amplification of transient energy in the counter-rotating regime, which aligned well with Coles’[31] experimental findings. It was discovered that the non-axisymmetric modes experience greater amplification and they could potentially trigger the subcritical transition to turbulence [19,36]. Maretzke et al.[37] further expanded on this perspective through numerical and analytical methods. The authors explored the transient energy growth across all TCF regimes and established a universal Re2/3 scaling. Additionally, it was established that the linear stability and transient growth remained un- affected by the ratio of cylinder rotation in the limits where there was no axial perturbation dependency.

However, there has been little attention regarding the application non-modal analysis to Stratified Taylor-Couette flow (STCF). Thus, the purpose of this work, is to extend the experimental work of Cole in the direction of thermal convection by means of non-modal analysis. We superimpose radial heating on the reported configuration of Coles[31] as given by Meseguer[36] and proceed with the investigation of the subcritical phenomena for the regime of the counter rotating flows.

This article is concerned with optimal transient growth in counter rotating stratified Taylor-Couette flow with non-axisymmetric perturbations. A brief description of the governing equations of the Taylor-Couette problem is presented in Section I. In Section II, the derived model is linearized. Section III describes the non-modal analysis method for computing the optimal energy growth. In section IV, a detailed discussion is presented for the results obtain in our investigation. Finally the paper is concluded in section V.

2. Mathematical Formulation

2.1. Model

We examine the behaviour of an incompressible fluid with kinematic viscosity, µ and a density, ρ = ρ_o + ρ′, which is contained between two concentric cylinders of infinite height. The inner cylinder has a radius, r_i and rotates at an angular velocity, Ω_i > 0, while the outer cylinder has a radius, r_o and rotates at an angular velocity, Ω_o < 0. Additionally, we assume a negative temperature gradient with inner temperature, T_i = $\stackrel{-}{T}$ + ∆T/2 and outer temperature, T_o = $\stackrel{-}{T}$ − ∆T/2, as depicted in Figure 1, where $\stackrel{-}{T}$ is the ambient temperature and ∆T is the temperature difference between the two cylinders. Thus, the model is governed by the incompressible Navier-Stokes equations for rapidly rotating flows, as described by Lopez et al. [38,39]:

(1)

(2)

∂_t T + (v · ∇)T = κ∇² T,

(3)

The boundary conditions are given as:

(4)

where µ and κ are the viscosity and thermal diffusivity of the fluid respectively; Π = p + ρ_o Φ, is composed of the gradient terms: pressure p, and constant gravitational potential ρ_o Φ. Equation (1) is defined in the stationary reference frame and the last term (ρ′ (v · ∇)v) describes the centrifugal buoyancy contribution of the flow. This term is necessary in order to accurately account for strong vortex dynamics or flows with very high rotating walls.

2.2. Boussinesq Approximation

We suppose the system is defined by a cylindrical coordinate system (r, θ, z). Thus, we proceed with the Boussinessq assumption by ignoring density variation in the flow, and keep only density contributions from the gravitational and centrifugal terms. Also, we assume a uniform gravitational acceleration, g in the vertical direction, z. Thus we have:

ρ = ρ_o (1 − βT)

(5)

But ρ = ρ_o + ρ′ based on the assumption in defining (1) given by Lopez et al. [38]. Thus we have:

ρ′ = −ρ_o βT

(6)

Now applying (6) to the buoyancy term in (1) and using Φ = gz, we have:

(7)

where β = −(1/ρ_o)∂ρ/∂T|ρ=ρ_o is the coefficient of thermal expansion.

2.3. Nondimensionalization

In this section we proceed by converting Equations (2), (3) and (7) to dimensionless form by using ∆T, gap-width d = r_o − r_i, d²/ν, (ν/d)² as the temperature, length, viscous time, and reduced pressure p/ρ_o scaling factors respectively. The inner r_i and outer r_o radius are expressed in dimensionless form as η/(1 − η) and 1/(1 − η) respectively. With these scaling factors we obtained new dimensionless parameters listed in Table 1 are obtained. Also, we introduce a dummy variable, Υ ∈ {0, 1} into the equation:

(8)

(9)

(10)

If Υ is 1, then (8) describes the STCF and if it is 0, then it becomes TCF without stratification. Thus, we use these parameters and the dimensionless Equations (8)–(10) in subsequent sections for our analysis and discussion.

2.4. Basic Equations

Given the nondimensional equations, let the velocity be expressed as v = ue_r + ve_θ + we_z in cylindrical coordinates, where u, v and w are the radial, azimuthal and axial components of the velocity, respectively, and e_r, e_θ and e_z are the unit vectors in the r, θ and z directions. The basic velocity U, becomes:

U = U_be_r + V_be_θ + W_be_z

(11)

and the basic components of the velocity U_b, V_b and W_b, corresponds to the respective radial, azimuthal and axial basic profiles. Thus we proceed by deriving expression for the basic profiles by assuming that, U is independent of time t, azimuthal θ and axial z directions. In addition, in order to have a constant pressure gradient in the axial direction, we impose the zero mass flux constraint. That is,

(12)

and thus, with these assumptions we obtain the following analytical solutions:

(13)

(14)

(15)

(16)

where V_b, W_b and T_b remain the same solutions introduced in related literature. The constants A, B and C are given by the expressions

(17)

and

(18)

4. Numerical Results

The amplification factor, often denoted as G₀, is a fundamental concept in the study of transient growth in fluid dynamics. It quantifies the maximum amplification of perturba- tions or disturbances in a given flow system over a certain time interval. Transient growth refers to the temporary amplification of perturbations in a flow before they eventually decay or become insignificant. This phenomenon is particularly relevant in subcritical transition to turbulence, where small disturbances can trigger the onset of turbulent behaviour. The amplification factor, G₀, represents the maximum energy amplification that can be achieved by the optimal initial perturbations within a specified time frame. It characterizes the efficiency of energy transfer from the mean flow to the perturbations. A larger value of G₀ indicates a higher potential for transient growth and the presence of strong amplification mechanisms in the flow. The computation of G₀, as defined in Appendix A, involves solving an eigenvalue problem or performing a non-modal analysis, depending on the nature of the flow and the mathematical framework employed. By determining the optimal initial conditions and analyzing the linearized equations governing the flow, G₀ can be evaluate. In addition, the structures and modes that contribute most significantly to the amplification can be identified. In our study, we employed well-established approaches from the literature to compute the amplification factor, G₀. These approaches have been widely used and validated in previous research, providing reliable and effective methods for evaluating the maximum amplification of disturbances in fluid flows. By utilizing these established techniques, we ensure the accuracy and robustness of our calculations, allowing us to gain valuable insights into the transient growth and stability characteristics of the flow system under investigation [5,40,41].

The configuration is chosen so that our results can compare well with earlier studies of Taylor-Couette flow (TCF) without stratification [35,36,37]. Thereafter, we apply stratification to the problem to investigate the behaviour of the transient energy growth of the flow . Thus, we continue with equivalent configuration used by Meseguer[36] for the study of energy transient growth in TCF problem. We choose fixed values of η = 0.881, ϵ = 0.067, Pr = 68, various range of Gr and wavenumber pairs (n ∈ [0, 15], k ∈ [0, 15]). In addition, we consider specific counter rotating pairs of values for Re_i and Re_o from the set of significant pairs reported by the author for TCF. Furthermore, for proper justification of this investigation, we consider the symmetry introduce due to the periodic assumption made for the azimuthal (i.e. the SO(2)-symmetry because of the invariant of the system in respect of the azimuthal rotation about the center axis) and axial (i.e. the O(2)- symmetry because of the invariant of the system in respect of the reflections and translation of the axial axis) coordinate directions.

Due to the fact that no experimental work exists for the choice of configuration to which we can compare our results, we have introduced a parameter Υ in the governing dimensionless equation with values ranging from 0 to 1. The value of Υ is set to 0 in order to compare our findings with already existing results obtained for TCF (refs). The results as indicated in Figure 2 are in perfect agreement with the results reported by Hristova et al. [35], Meseguer [36], and recently by works of Maretzke et al. [37]. In addition, we verify that the eigenvalues of STCF is in perfect agreement with Lopez [38], and McFadden et al. [38].

The nondimensional scheme we used, is similar to Meseguer[36] and Maretzke et al.[37], but is slightly different from the scales used by Hristova et al.[35]. Thus, we made the conversions Re_i = 2Re_i and Re_o = 2Re_o /η in computing the results with those reported by Hristova et al.[35] and as done by others [36,37]. This is because, the authors used a scale of d/2 to non-dimensionalize the length and fix the angular rotation speed in defining the problem.

Meseguer[36] and Maretzke et al. [37] reported that close to the region where Re_i = ηRe_o (i.e. the cylinders rotate at the same speed) no transient growth is observed. However, it is observed that the situation is different when the temperature increases beyond a certain threshold. There exists an amplification of the perturbation at specific times for different relative counter-rotating speeds in the presence of temperature variations. We continue by applying radial heating to the TCF problem and investigate the effect on the thermal stratification for various Grashof, Gr, numbers with specific counter rotating pairs Re_i, Re_o and wavenumbers. Figure 3 shows the amplification factor G against t for the four different configurations C₁ to C₄ for various values of Gr. We find that G has a minimum around Gr=3250 for configuration C₁, around Gr=4800 for configuration C₂, around Gr=5750 for configuration C₃, and around Gr=7150 for configuration C₄. These results are verified explicitly in Figure 4b. Figure 4 shows that as the temperature increases, the amplification of the perturbation monotonically decays, before reaching a turning point where it enters a new phase and begins increasing monotonically with temperature.

The growth rate shows a linear relationship with Gr in all configurations, as evident from the Figure 4a. The growth rate remains consistently low across all Gr values, indi- cating linear stability of the fluid within the considered range of flow and implying that buoyancy does not induce linear instability. Figure 4a clearly demonstrates an increase in the growth rate with higher Gr values, which aligns with the proportional relationship be- tween G₀ and Gr as illustrated. We observe in Figure 4b that initially, there is an observable downward trend in the linear decay of G₀ as Gr increases until it reaches a critical value Gr. Beyond this critical value, a linear increase in G₀ is observed as Gr continues to rise. By examining the decay phase boundaries, the order of G₀ for each configuration can be represented by the expression G₀(C₄) > G₀(C₃) > G₀(C₂) > G₀(C₁). Similarly, for the increasing phase, the order of G₀ can be characterized as G₀(C₄) < G₀(C₃) < G₀(C₂) < G₀(C₁).

An intriguing observation depicted in Figure 4b is the relationship between C₁ and the other configurations, namely C₂, C₃, and C₄, with respect to their ratio and the magnitude of their growth rates. Notably, there is a bigger difference in the order of growth values between C₁ and C₂ when compared to the difference between C₂ and C₃ across all Gr values. The consistency in the proportionality of their speed ratio suggests that their growth rates should exhibit similar differences in the same order of magnitude. This observation could be due to the influence of buoyancy-induced contributions captured in the amplification reflected by G₀.

The underlining behaviour of the fluid motion can be attributed to an internal sub-critical phenomena because of the induced thermal stratification on the dynamics of the disturbances.

During the decay phases of the amplification factor, it is evident that the combination of viscous and buoyancy influences the amplitude of the perturbations. In Figure 5, the amplification factor G is depicted to illustrate its growth as Gr increases. Each row in the sequence of plots represents the evolving behaviour of G as Gr increases. The first Gr = 2250, 3800, 3750, 5150 and last columns of each sequence represented by Gr = 4250, 5800, 7750, 9150 exhibit Gr values slightly distant from the turning point, Gr = 3250, 4800, 5750, 7150 showcasing a single peak with a certain amplitude. Additionally, the plots in the second and fourth columns demonstrate two distinct behaviors in the evolution. Notably, the second column plots display the emergence of a new crest on the wave front. This new crest continues to grow until it surpasses the initial crest in amplitude, eventually becoming the sole peak after the turning point for sufficiently high Gr values as depicted by the sequence. The third column has used a value of Gr where both crescents exhibit similar amplitudes. The initial crest gradually merges into the latter one, and with further increases in Gr, the former completely disappears due to induced buoyancy disturbances.

In Figure 6 we illustrate contour plots of the amplified perturbations in the radial velocity $\breve{u}$ at the value of Gr when G = G₀. The sequence of contour plots follows that of Figure 5 for C₁ to C₄. Notice t₋₂, t₋₁, t₀, t₁ and t₂ follow an arithmetic progression, where t₀ is defined as the time when G = G₀. We see in Figure 6 that at G = G₀ we have a reversal in the rotational direction of the perturbed flow.

5. Conclusion

Over the past twenty years, researchers have widely utilized non-normal analysis to investigate the linear transient characteristics of wall bounded flows, particularly shear flows. However, there has been a relatively limited focus on applying this approach to examine the behaviour of Stratified Taylor-Couette Flow (STCF).

This study aims to explore the dynamics of perturbations under different tempera- ture conditions and analyze their patterns of amplification. The research emphasizes the correlation between various flow configurations, highlighting their similarity in transient dynamics despite differing speed ratios. Additionally, it investigates the subcritical effects of thermal stratification on the dynamics of the disturbances. The interplay between viscous and buoyancy effects, counteracted by strong centrifugal forces, is found to influence the growth and decay of the amplification factor. As the temperature increases, beyond a turning point, buoyancy forces become dominant, leading to a linear increase in the ampli- fication factor. The findings of this study reveal that incorporating temperature into the analysis significantly alters the expected outcomes, deviating significantly from the results obtained by Meseguer[36] and Coles[31]. This supports the notion that buoyancy, induced by the stratification of temperature variation, plays a significant role. The transient analysis suggests the presence of complex dynamical phenomena specific to transient dynamics, which cannot be captured by solely focusing on the growth rate of the eigenvalue.

However, it is important to note that the occurrence of this phenomenon and its de- tectability through physical laboratory experiments are yet to be determined. Experimental verification would be beneficial to confirm these findings.