A. Liakopoulos

Stability of two-layer poiseuille flow of Carreau-Yasuda and Bingham-like fluids

Abstract In this paper we present the linear stability analysis of the interface between two non-Newtonian inelastic fluids in a straight channel driven by a pressure gradient. Two rheological models of non-Newtonian behavior are studied: (a) Bingham-like fluids and (b) Carreau-Yasuda fluids. For each rheological model, the linearized equations describing the evolution of small two-dimensional disturbances are derived and the stability problem is formulated as an eigenvalue problem for a set of ordinary differential equations of the Orr-Sommerfeld type. Discretization is performed using a pseudospectral technique based on Chebyshev polynomial expansions. The resulting generalized matrix eigenvalue problem is solved using the QZ algorithm. The results on the onset of instability are presented in the form of stability maps for a range of zero-shear-rate viscosity ratios, thickness ratios, power-law constants, material time constants, Yasuda constants, apparent yield stresses and stress growth exponents. Increasing the stress growth exponents or apparent yield stresses of the viscoplastic fluids has a stabilizing effect on the interface. For shear thinning fluids, increasing the zero-shear-rate viscosity ratio or shear thinning of the fluids destabilizes the interface. The effect of other parameters can be stabilizing or destabilizing depending on the flow configuration and wavelength.

A reduced dynamical model of convective flows in tall laterally heated cavities

Proper orthogonal decomposition (the Karhunen–Loève expansion) is applied to convective flows in a tall differentially heated cavity. Empirical spatial eigenfunctions are computed from a multicellular solution at supercritical conditions beyond the first Hopf bifurcation. No assumption of periodicity is made, and the computed velocity and temperature eigenfunctions are found to be centro‐symmetric. A low-dimensional model for the dynamical behaviour is then constructed using Galerkin projection. The reduced model successfully predicts the first Hopf bifurcation of the multicellular flow. Results determined from the low-order model are found to be in qualitative agreement with known properties of the full system even at conditions far from criticality.

The application of complex network time series analysis in turbulent heated jets

In the present study, we applied the methodology of the complex network-based time series analysis to experimental temperature time series from a vertical turbulent heated jet. More specifically, we approach the hydrodynamic problem of discriminating time series corresponding to various regions relative to the jet axis, i.e., time series corresponding to regions that are close to the jet axis from time series originating at regions with a different dynamical regime based on the constructed network properties. Applying the transformation phase space method (k nearest neighbors) and also the visibility algorithm, we transformed time series into networks and evaluated the topological properties of the networks such as degree distribution, average path length, diameter, modularity, and clustering coefficient. The results show that the complex network approach allows distinguishing, identifying, and exploring in detail various dynamical regions of the jet flow, and associate it to the corresponding physical behavior. In addition, in order to reject the hypothesis that the studied networks originate from a stochastic process, we generated random network and we compared their statistical properties with that originating from the experimental data. As far as the efficiency of the two methods for network construction is concerned, we conclude that both methodologies lead to network properties that present almost the same qualitative behavior and allow us to reveal the underlying system dynamics.

Quantitative Imaging in Proper Orthogonal Decomposition of Flow Past a Delta Wing

Snapshot proper orthogonal decomposition is used to characterize the unsteady e ow past a delta wing, focusing on the structure of the leading-edge vortices. The decomposition is applied to two-dimensional velocity e elds obtained via a laser-scanning version of high-image-density particle image velocimetry to extract the most coherent structures in the e ow. Data are analyzed for cases involving both a stationary wing and a wing undergoing harmonic oscillations in roll. In each case, the velocity e elds are reconstructed as a truncated series expansion in terms of the computed eigenfunctions from which the corresponding contours of constant vorticity and sectional streamline patterns are calculated. Comparison with the original data demonstrates that the analysis provides an accurate representation of the velocity e elds while eliminating extraneous small-scale features. Inclusion of as few astwo eigenfunctionsin thereconstruction series reproduces the largest-scalefeaturesoftheleading-edge vortices, whereas the inclusion of half of the total number of eigenfunctions produces a reconstructed e eld that captures the majority of the e ow features. By appropriately combining the spatial and temporal components of the proper orthogonal decomposition analysis, one obtains the dynamical structures that evolve in both space and time, such as the e uctuations in the location of vortex breakdown. Nomenclature ak = modal amplitudes for velocity b = span C = correlation matrix co = centerline chord dI = interrogation window size E = mean total e uctuation energy M = total number of snapshots N = number of modes retained in Eq. (4) T = period of oscillation t = time U1 = freestream velocity u = x component of velocity V = velocity vector v = y component of velocity xvb = location of vortex breakdown, measured along wing centerline from apex x; y = coordinates ®k = kth eigenvector of the correlation matrix 1t = time interval between snapshots · = reduced frequency, .ob/=2U1T ¸k = kth eigenvalue of the correlation matrix 8 = roll angle Ak = kth empirical velocity eigenfunction N = time-averaged quantities 0 = e uctuation quantities

The role of variable viscosity in the stability of channel flow

Abstract The stability of plane Poiseuille flow is studied for liquids exhibiting exponential viscosity-temperature dependence. In contrast to previously published studies, viscosity and temperature fluctuations are included in the formulation. Equations describing the evolution of small, two-dimensional disturbances are derived and the stability problem is formulated as an eigenvalue problem for a set of two ordinary differential equations. A Chebyshev collocation discretization method leads to a generalized matrix eigenvalue problem which is solved by the QZ algorithm. It is found that an imposed wall temperature difference, Δ T − , is always destabilizing. The instability region in the wavenumber-Reynolds number plane grows considerably as Δ T − increases. The influence of Prandtl number, temperature fluctuations and viscosity fluctuations on the flow stability/instability is small. However, their influence on the margin of stability for small wavenumbers is appreciable.

Vortex Buffeting of Aircraft Tail: Interpretation via Proper Orthogonal Decomposition

The proper-orthogonal-decomposition method is applied to particle-image-velocimetry (PIV) data to extract the most energetic flow structures of vortex-tail interaction. The reconstructed flowfields, in conjunction with patterns of vorticity and streamline topology, are compared with the original PIV data on a crossflow plane. The reconstructed flowfields using 5 and 10 eigenfunctions can predict the largest-scale features of the original flowfield. However, the smaller-scale flow structures, which are evident in the original PIV image, are lost

Reduced dynamical models of nonisothermal transitional grooved-channel flow

Reduced dynamical models are derived for transitional flow and heat transfer in a periodically grooved channel. The full governing partial differential equations are solved by a spectral element method. Spontaneously oscillatory solutions are computed for Reynolds number Re⩾300 and proper orthogonal decomposition is used to extract the empirical eigenfunctions at Re=430, 750, 1050, and Pr=0.71. In each case, the organized spatio-temporal structures of the thermofluid system are identified, and their dependence on Reynolds number is discussed. Low-dimensional models are obtained for Re=430, 750, and 1050 using the computed empirical eigenfunctions as basis functions and applying Galerkin’s method. At least four eigenmodes for each field variable are required to predict stable, self-sustained oscillations of correct amplitude at “design” conditions. Retaining more than six eigenmodes may reduce the accuracy of the low-order models due to noise introduced by the low-energy high order eigenmodes. The low-orde...

On a class of compressible laminar boundary-layer flows and the solution behaviour near separation

A class of compressible laminar boundary-layer flows subject to adverse pressure gradients of different magnitude is studied using a finite-element-differential method in which the assumed solutions are represented by classical cubic spline functions. The numerical integration process for the reduced initial-value problem has been carried out directly to a t least one integration step upstream of the separation point, and very accurate numerical results have been obtained for a large number of integration steps extremely close to separation. The skin-friction and heat-transfer coefficients for nearly zero-heat-transfer, cooled-wall and heated-wall cases, computed under the assumption of constant Prandtl number Pr = 1 .O as well as Pr = 0.72, have clearly exhibited the same distinctive behaviour near separation. It is deduced that Buckmaster’s series expansions for the solution near separation, derived on the assumptions of cooled wall and Pr = 1.0, are valid for all the cases considered. By matching the numerical results with Buckmaster’s expansions, accurate distributions of skin friction and heat transfer have been obtained up to the separation point. Moreover, the importance of Prandtl number on the solution is evidenced from the numerical results presented.

The effect of variable viscosity on the interfacial stability of two‐layer Poiseuille flow

In this paper the linear stability analysis of the interface between two Newtonian liquids with temperature‐dependent viscosity in plane Poiseuille flow is presented. A piecewise linear temperature profile is considered. The linearized equations describing the evolution of small, two‐dimensional disturbances are derived and the stability problem is formulated as an eigenvalue problem for a set of ordinary differential equations. The continuous eigenvalue problem is solved numerically by a pseudospectral method based on Chebyshev polynomial expansions. The method leads to a generalized matrix eigenvalue problem, which is solved by the QZ algorithm. Results on the onset of instability are presented in the form of stability maps for a range of thickness ratios, disturbance wave numbers, imposed temperature differences, constant‐temperature viscosity ratios, thermal conductivity ratios, and Reynolds numbers. Increasing the imposed temperature difference, constant‐temperature viscosity ratio, or Reynolds numbe...

A modeling approach to transitional channel flow

Abstract Low-dimensional dynamical models of transitional flow in a periodically grooved channel are numerically obtained. The governing partial differential equations (continuity and Navier–Stokes equations) with appropriate boundary conditions are solved by a spectral element method for Reynolds number Re=430. The method of empirical eigenfunctions (proper orthogonal decomposition) is then used to extract the most energetic velocity eigenmodes, enabling us to represent the velocity field in an optimal way. The eigenfunctions enable us to identify the spatio-temporal (coherent) structures of the flow as travelling waves, and to explain the related flow dynamics. Using the computed eigenfunctions as basis functions in a truncated series representation of the velocity field, low-dimensional models are obtained by Galerkin projection. The reduced systems, consisting of few non-linear ordinary differential equations, are solved using a fourth-order Runge–Kutta method. It is found that the temporal evolution of the most energetic modes calculated using the reduced models are in good agreement with the full model results. For the modes of lesser energy, low-dimensional models predict typically slightly larger amplitude oscillations than the full model. For the slightly supercritical flow at hand, reduced models require at least four modes (capturing about 99% of the total flow energy). This is the smallest set of modes capable of predicting stable, self-sustained oscillations with correct amplitude and frequency. POD-based low-dimensional dynamical models considerably reduce the computational time and power required to simulate transitional open flow systems.