ETC2019 17th European Turbulence Conference Submission

16:15 Stratified Flows 2

16:15 15 mins #175	REVISITING BOLGIANO-OBUKHOV SCALING FOR STABLY STRATIFIED TURBULENCE Shadab Alam, Anirban Guha, Mahendra K Verma Abstract: According to Bolgiano and Obukhov, buoyancy force in moderately stably stratified turbulence converts kinetic energy into potential energy. Bolgiano-Obukhov (BO) phenomenology describes that the kinetic energy flux in the inertial range decreases with the wavenumber k as k^{-4/5}, however, the potential energy flux in the inertial range is assumed to be a constant. As a result, the energy spectra in the inertial range show a dual scaling. The kinetic energy spectrum and potential energy spectrum, respectively, show k^{-11/5} and k^{-7/5} scaling for k less than the Bolgiano wavenumber. For k greater than Bolgiano wavenumber, both the spectra follow k^{-5/3} scaling. The k^{-5/3} scaling, akin to passive scalar turbulence, is a direct consequence of the assumption that energy supply rate by buoyancy is insignificant for k greater than the Bolgiano wavenumber. The crucial assumption of constant potential energy flux made in BO phenomenology needs a closer examination. So, we revisit the phenomenology, and use a more rigorous approach of the constancy of the sum of kinetic and potential energy fluxes, resulting from the conservation of energy in the inviscid limit. With simple numerical computation as well as theoretical analysis, we demonstrate that for k > O(1), the potential energy flux is approximately constant and the kinetic energy flux follows k^{-4/5}. The kinetic and the potential energy spectra, respectively, follow k^{-11/5} and k^{-7/5} scaling, and that no transition to passive scalar turbulence scaling k^{-5/3} occurs. This is because the velocity field at small scales is too weak to establish a constant kinetic energy flux as in passive scalar turbulence. Interestingly, for k < O(1), the potential energy spectrum follows k^{4/3} and the kinetic energy flux is approximately constant. The kinetic and the potential energy spectra, respectively, follow k^{-5/3} and k^{-1/3} scaling. Note however that k = 1 corresponds to 1/L. Hence, k << 1 is possible in stably stratified turbulence when the transverse length scale is much larger than the vertical scale (L), however, this regime may lead to inverse cascade of kinetic energy as in two-dimensional and quasi-two-dimensional turbulence. This prediction needs to be tested in numerical simulations. Our findings may have important implications in the modelling of stably stratified turbulence.
16:30 15 mins #279	Decaying turbulence in a stratified fluid generated by a high-Prandtl-number scalar Hideshi Hanazaki, Shinya Okino Abstract: Decaying turbulence in density-stratified fluid is investigated by the direct numerical simulation with the Prandtl number up to 700. In isotropic turbulence, it has been known that the fluctuations of a passive scalar with the Prandtl number larger than unity cascade below the Kolmogorov scale with the k^{-1} spectrum, down to the Batchelor scale[1]. The phenomenon can be observed also for a buoyant scalar in stratified fluids, in its early development. For example, the early-time spectra in decaying stratified turbulence (figure 1a) shows that the scalar of Pr =70 has the k^{-1} spectrum near the Kolmogorov scale. This can be observed as long as the Ozmidov scale, which decreases with time, is larger than the increasing Kolmogorov scale, so that the buoyancy is not effective below the Kolmogorov scale. Once the Ozmidov scale becomes smaller than the Kolmogorov scale, the velocity field near the Kolmogorov scale begins to show a strong anisotropy dominated by the vertically sheared horizontal flow. This flow reduces the vertical scale of the density fluctuations, and generates the large-scale pancake structures as observed in figure 1(b). The analysis similar to Batchelor[1] indeed shows that the vertically sheared horizontal flow reduces the vertical scale of density fluctuations, without affecting the horizontal scale. Figure 1(b) also suggests that the fluctuations much smaller than the Kolmogorov scale survive at very high $Pr(=700)$, while the fluctuations near the Kolmogorov scale reduce significantly. This could be explained by the energy conversion from the potential energy to the kinetic energy by the large counter-gradient density flux near the Kolmogorov scale.
16:45 15 mins #293	Kelvin-Helmholtz instability above Richardson number 1/4 Jeremy Parker, Colm-cille Caulfield, Rich Kerswell Abstract: Kelvin-Helmholtz-type instabilities are believed to be responsible for much of the turbulent mixing in the oceans, through the breaking of internal gravity waves in stably stratified environments. In an idealised flow, if the density stratification, as quantified by the Richardson number, is sufficiently strong, no instabilities can exist, as shown by the Miles-Howard theorem. However, this does not preclude the possibility of finite-amplitude perturbations leading to turbulence through non-linear effects. We study the dynamical system of a stratified mixing layer at finite Reynolds number and unity Prandtl number. We consider a hyperbolic tangent background velocity profile and both hyperbolic tangent and uniform background density stratifications. The system is forced in such a way that these background profiles are a steady solution of the governing equations. As is well-known, if the minimum Richardson number of the flow,Rim, is less than a certain critical value Ric, the flow is linearly unstable to Kelvin-Helmholtz instability. Using Newton-Krylov iteration, we find steady, two-dimensional, finite amplitude elliptical vortex structures, i.e. 'Kelvin-Helmholtz billows', existing above Ric. Bifurcation diagrams are produced using branch continuation, and we explore how these diagrams change with varying Reynolds number. In particular, we find that finite amplitude Kelvin-Helmholtz billows exist at Rim>1/4, where the flow is linearly stable by the Miles-Howard theorem, when the Reynolds number is sufficiently high. For a uniform background stratification, we give a simple explanation of the dynamical system, showing the dynamics can be understood on a two-dimensional subspace of the whole state space, and demonstrate the cases in which the system is bistable. In the case of a hyperbolic tangent background density, we show that the artificial forcing complicates this picture.
17:00 15 mins #399	Regime transitions and energetics of sustained stratified shear flows Adrien Lefauve, Jamie Partridge, Paul Linden Abstract: We describe the long-term dynamics of sustained stratified shear flows in the laboratory. The Stratified Inclined Duct (SID) experiment sets up a two-layer exchange flow in an inclined duct connecting two reservoirs containing salt solutions of different densities. This flow is primarily characterised by two non-dimensional parameters: the tilt angle of the duct with respect to the horizontal, θ (a few degrees at most), and the Reynolds number Re, an input parameter based on the density difference driving the flow. The flow can be sustained with constant forcing over arbitrarily long times and exhibits a wealth of dynamical behaviours representative of geophysically-relevant sustained stratified shear flows. Varying θ and Re leads to four qualitatively different regimes: laminar flow; mostly laminar flow with finite-amplitude, travelling Holmboe waves; spatio-temporally intermittent turbulence with substantial interfacial mixing; and sustained, vigorous interfacial turbulence (Meyer & Linden, J. Fluid Mech., vol. 753, 2014, pp. 242-253). In this paper, we seek to explain the scaling of the transitions between flow regimes in the two-dimensional plane of input parameters (θ,Re). We improve upon previous studies of this problem by providing a firm physical basis and non-dimensional scaling laws that are mutually consistent and in good agreement with the empirical transition curves we inferred from 360 experiments spanning θ∈[−1∘,6∘] and Re∈[300,5000]. To do so, we employ state-of-the-art simultaneous volumetric measurements of the density field and the three-component velocity field, and analyse these experimental data using time- and volume-averaged potential and kinetic energy budgets. We show that regime transitions are caused by an increase in the non-dimensional time- and volume-averaged kinetic energy dissipation within the duct, which scales with θRe at high enough angles. As the power input scaling with θRe is increased above zero, the two-dimensional, parallel-flow dissipation (power output) increases to close the budget through an increase in the magnitude of the exchange flow, incidentally triggering Holmboe waves above a certain threshold in interfacial shear. However, once the hydraulic limit of two-layer exchange flows is reached, two-dimensional dissipation plateaus and three-dimensional dissipation at small scales (turbulence) takes over, first intermittently, and then steadily, in order to close the budget and follow the θRe scaling. This general understanding of regime transitions and energetics in the SID experiment may serve as a basis for the study of more complex sustained stratified shear flows found in the natural environment.
17:15 15 mins #314	SENSITIZATION OF EDDY-VISCOSITY MODELS TO BUOYANCY EFFECTS FOR PREDICTING NATURAL CONVECTION FLOWS Syed Mohd Saad Jameel, Remi Manceau, Vincent Herbert Abstract: The influence of buoyancy on turbulent flows is significant in many industrial applications, in particular in the automotive industry. For instance, CFD is routinely used for dimensioning the underhood compartment, using commercial packages based on eddy-viscosity turbulence models. Although these models correctly reproduce the flow and heat transfer at cruising speed, they are not reliable in phases where natural convection dominates, i.e., when the car is stopped, mainly because buoyancy-turbulence interactions are not accounted for in a comprehensive manner. The objective of the present work is to introduce the mechanisms involved in this interaction in eddy-viscosity models in order to avoid the recourse to costly wind-tunnel experiments. The study is focused on simple-geometry flows for which the influence of the different physical processes can be isolated and the corresponding terms in the equations are available in DNS databases. Moreover, in order to ensure that the modifications introduced to account for buoyancy effects do not deteriorate predictions in the absence of buoyancy, the study encompasses forced, mixed and natural convection regimes, covering a range of Rayleigh numbers representative of underhood flow configurations: channel flows in forced, mixed and natural convection regimes, up to Ra = 1.7E+07 [2] and differentially heated cavities up to Ra = 10E+11 [3]. Buoyancy effects are commonly introduced by adding source terms to the transport equations for the turbulent energy and the (specific) dissipation rate, in order to reproduce the influence of stratification on the dynamics of turbulence. However, this approach is ineffective in weakly stratified buoyant flows, where the temperature gradient is mainly normal to the gravity vector, as a consequence of the use of an isotropic thermal diffusivity (SGDH) to model the turbulent heat fluxes. However, the present work shows that introducing an anisotropic diffusivity (GGDH) only marginally improves the predictions. Indeed, the use of the Boussinesq constitutive relation to model the Reynolds stresses in eddy-viscosity models does not correctly represent the subtle coupling between the turbulent dynamics and the turbulent heat fluxes: the influence of buoyancy on the anisotropy of turbulence must be accounted for, in order to correctly represent the anisotropic diffusivity, which is crucial to reproduce the turbulent heat fluxes and the production mechanisms in the turbulent energy and dissipation equations. The main contribution of this work is thus the drastic improvement of the reproduction of the influence of buoyancy on turbulence in the case of weakly stratified flows in the natural convection regime, through the modification of both the turbulent heat flux model and the constitutive relation for the Reynolds stress. The former can be modeled using either an anisotropic diffusivity (GGDH) or the more sophisticate Algebraic Flux Model (AFM), and the latter is modified by introducing a buoyancy extension in the form [1]. The models described above have been implemented in the open-source CFD package Code_Saturne, in combination with different types of eddy-viscosity models (k-epsilon, k-omega, BL-v2/k). Detailed comparisons with DNS of the results obtained using the different hypotheses show that all the forced, mixed and natural convection regimes can be reproduced in a satisfactory manner by combining the GGDH and the buoyancy-sensitized constitutive relation (1). In particular, in natural convection, the mean velocity profiles in boundary layers close to the vertical walls are drastically improved. Detailed comparison of all the significant quantities will be presented, as well as the analysis of the influence of the different hypotheses on the reproduction of the coupling of the dynamics and heat transfer mechanisms. The results are thus very promising and this work paves the way to the improvement of CFD capabilities for the design of underhood compartment of automobiles, and, in general, for industrial configurations in which buoyancy has a significant influence.
17:30 15 mins #78	Subcritical and supercritical transitions for stratified fluid in a nearly semicircular pool Abhishek Kumar Abstract: See the attached file.
17:45 15 mins #507	Sudden transition from non-swirling to swirling axisymmetric turbulence Wouter Bos, Zecong Qin, Aurore Naso Abstract: Strictly axisymmetric turbulence, i.e. turbulence governed by the Navier-Stokes equations modified such that the flow is invariant in the azimuthal direction, is a system intermediate between two- and three-dimensional turbulence. We show by simulations the remarkable feature of the system, that the toroidal energy remains zero even when the toroidal forcing is non-zero. Only when the forcing of both components becomes comparable, the flow abrubtly transitions towards a swirling state. We then derive a statistical model for this behaviour, starting from the axisymmetric Navier-Stokes equations, which very accurately reproduces the transition between the two flow states.
18:00 15 mins #366	Asymptotic dynamics of high dynamic range stratified turbulence Colm-cille Caulfield, Gavin Portwood, Steve de Bruyn Kops Abstract: Direct numerical simulations of homogeneous sheared and stably stratified turbulence are considered to probe the asymptotic high-dynamic range regime\cite{gargett84,shih05}. We consider statistically stationary configurations of the flow that span three decades in dynamic range defined by the separation between the Ozmidov length scale, $L_O=\sqrt{\epsilon/N^3}$, and the Kolmogorov length scale, $L_K=(\nu^3/\epsilon)^{1/4}$, up to $\Gn\equiv (L_O/L_K)^{4/3}=\epsilon/(\nu N^2) \sim O(1000)$, where $\epsilon$ is the mean turbulent kinetic energy dissipation rate, $\nu$ is the kinematic viscosity, and $N$ is the buoyancy frequency. We isolate the effects of $\Gn$, particularly on irreversible mixing, from the effects of other flow parameters of stratified and sheared turbulence, by allowing the forced flow to adjust towards stationarity. Specifically, we evaluate the influence of dynamic range independent of initial conditions, finding that for a wide variety of simulations with $36 \leq \Gn\leq 900$, the flow always evolves such that $Ri \equiv N^2/S^2 \simeq 0.16$, and $Fr \equiv \epsilon/(N E_k) \simeq 0.5$, where $S$ is the (imposed) shear, and $E_k$ is the turbulent kinetic energy. We present evidence that the flow approaches an asymptotic state for $\Gn\gtrapprox 300$, characterized both by an asymptotic partitioning between the potential and kinetic energies and by the approach of components of the dissipation rate to their expected values under the assumption of isotropy. As $\Gn$ increases above 100, there is a slight decrease in the turbulent flux coefficient $\Gamma=\chi/\epsilon$, as shown in figure \ref{fig1}a, where $\chi$ is the dissipation rate of buoyancy variance. However, for this flow, there is no evidence of the commonly suggested $\Gamma \propto \Gn^{-1/2}$ (dashed line) dependence when $100 \leq \Gn \leq 1000$. Indeed, $\Gamma$ remains very close to the upper bound $\Gamma \lesssim 0.2$ proposed in the classic Osborn parameterization\cite{osborn80}, while the turbulent Prandtl number $Pr_T \equiv \kappa_M/\kappa_T = (P/S^2)/(B/N^2) \simeq 1$ for all values of $\Gn$, as shown in figure \ref{fig1}b, where $P$ is the turbulence production and $B$ is the (vertical) buoyancy flux.