Sanjeev Sanghi

Mode interaction models for near-wall turbulence

Intermittent bursting events, similar to those characterizing the dynamics of near-wall turbulence, have been observed in a low-dimensional dynamical model (Aubry et al. 1988) built from eigenfunctions of the proper orthogonal decomposition (Lumley 1967). In the present work, we investigate the persistency of the intermittent behaviour in higher - but still of relatively low-dimensional dynamical systems. In particular, streamwise variations which were not accounted for in an explicit way in Aubry et al.s model are now considered. Intermittent behaviour persists but can be of a different nature. Specifically, the non-zero streamwise modes become excited during the eruptive events so that rolls burst downstream into smaller scales. When structures have a finite length, they travel at a convection speed approximately equal to the mean velocity at the top of the layer ( y + ≈ 40). In all cases, intermittency seems to be due to homoclinic cycles connecting hyperbolic fixed points or more complex (apparently chaotic) limit sets. While these sets lie in the zero streamwise modes invariant subspace, the connecting orbits consist of nonzero streamwise modes travelling downstream. Chaotic limit sets connected by quasi-travelling waves have also been observed in a spatio-temporal chaotic regime of the Kuramoto–Sivashinsky equation (Aubry & Lian 1992a). When the limit sets lose their steadiness, the elongated rolls become randomly active, as they probably are in the real flow. A coherent structure study in our resulting flow fields is performed in order to relate our findings to experimental observations. It is shown that streaks, streamwise rolls, horseshoe vortical structures and shear layers, present in our models, are all connected to each other. Finally, criteria to determine a realistic value of the eddy viscosity parameter are developed.

Proper orthogonal decomposition and low-dimensional modelling of thermally driven two-dimensional flow in a horizontal rotating cylinder

A proper orthogonal decomposition (POD) analysis and low-dimensional modelling of thermally driven two-dimensional flow of air in a horizontal rotating cylinder, subject to the Boussinesq approximation, is considered. The problem is unsteady due to the harmonic nature of the gravitational buoyancy force with respect to the rotating observer and is characterized by four dimensionless numbers: gravitational Rayleigh number ( Ra g ), the rotational Rayleigh number ( Ra Ω), the Taylor number ( Ta ) and Prandtl number ( Pr ). The data for the POD analysis are obtained by numerical integration of the governing equations of mass, momentum and energy. The POD is applied to the computational data for Ra Ω varying in the range 10 2 –10 6 while Ra g and Pr are fixed at 10 5 and 0.71 respectively. The ratio of Ta to Ra Ω is fixed at 100 so that the results apply to physically realistic situations. A new criterion, in the form of appropriately defined error norms, for assessing the truncation error of the POD expansion is proposed. It is shown that these error norms reflect the accuracy of the POD-based reconstructions of a given data ensemble better than the widely employed average energy criterion. The translational symmetry in both space and time of the pair of modes having degenerate (equal) eigenvalues confirms the presence of travelling waves in the flow for several different Ra Ω values. The shifts in space and time of the structure of the degenerate modes are utilized to estimate the wave speeds in a given direction. The governing equations for the fluctuations are derived and low-dimensional models are constructed by employing a Galerkin procedure. For each of the five values of Ra Ω , the low-dimensional models yield accurate qualitative as well as quantitative behaviour of the system. Sufficient modes are included in the low-dimensional models so that the modelling of the unresolved scales of motion is not needed to stabilize their solution. Not more than 20 modes are required in the low-dimensional models to accurately model the system dynamics. The ability of low-dimensional models to accurately predict the system behaviour for a set of parameters different from those from which they were constructed is also examined.

The Dynamics of Two-Dimensional Buoyancy Driven Convection in a Horizontal Rotating Cylinder

The present study involves a numerical investigation of buoyancy induced two-dimensional fluid motion in a horizontal, circular, steadily rotating cylinder whose wall is subjected to a periodic distribution of temperature. The axis of rotation is perpendicular to gravity. The governing equations of mass, momentum and energy, for a frame rotating with the enclosure, subject to Boussinesq approximation, have been solved using the Finite Difference Method on a Cartesian colocated grid utilizing a semi-implicit pressure correction approach. The problem is characterized by four dimensionless parameters: (1) Gravitational Rayleigh number Ra g ; (2) Rotational Rayleigh number Ra Ω ; (3) Taylor number Ta; and (4) Prandtl number Pr. The investigations have been carried out for a fixed Pr=0.71 and a fixed Ra g =10 5 while Ra Ω is varied from 10 2 to 10 7

Lift-drag and flow structures associated with the “clap and fling” motion

The present study focuses on the analysis of the fluid dynamics associated with the flapping motion of finite-thickness wings. A two-dimensional numerical model for one and two-winged “clap and fling” stroke has been developed to probe the aerodynamics of insect flight. The influence of kinematic parameters such as the percentage overlap between translational and rotational phase ξ, the separation between two wings δ and Reynolds numbers Re on the evolvement of lift and drag has been investigated. In addition, the roles of the leading and trailing edge vortices on lift and drag in clap and fling type kinematics are highlighted. Based on a surrogate analysis, the overlap ratio ξ is identified as the most influential parameter in enhancing lift. On the other hand, with increase in separation δ, the reduction in drag is far more dominant than the decrease in lift. With an increase in Re (which ranges between 8 and 128), the mean drag coefficient decreases monotonously, whereas the mean lift coefficient decre...

New Scheme for the Computation of Compressible Flows

A new approach for the computation of unsteady compressible flows has been developed. The new scheme employs upwinding of the convective flux based on particle velocity and has been termed the particle velocity upwinding (PVU) scheme. The PVU scheme is an explicit two-step predictor-corrector scheme, in which the convective fluxes are evaluated on cell faces using a first-order upwinding method. The scheme is accurate and stable, giving solutions free from oscillations near the discontinuities without any explicit addition of artificial viscosity. The PVU scheme has an edge over state-of-the-art high-resolution schemes in terms of simplicity of implementation in multidimensional flows and problems involving complex domains. The numerical scheme is validated for both Euler and Navier-Stokes equations. Furthermore, the PVU scheme is used to investigate laminar supersonic viscous flow over a forward-facing step. The results are obtained for M ∞ = 1.5-3.5 in steps of 0.5 and for Re ∞ = 10 4 . Step heights H s of 10 and 20% of the characteristic length of the problem are considered. The effect of step height and the incoming freestream Mach number on the spatial flow structure and on the important design parameters such as wall pressure, skin friction, heat transfer, and length of separated region are investigated.

Natural convection in a bottom heated horizontal cylinder

In this work a numerical investigation of two-dimensional steady and unsteady natural convection in a circular enclosure whose lower half is nonuniformly heated and upper half is maintained at a constant lower temperature has been carried out. An explicit finite difference method on a nonstaggered rectangular grid is used to solve the momentum and energy equations subject to Boussinesq approximation. The study is carried out for a range of Rayleigh number (Ra) varying between 102 and 106 at a fixed Prandtl number (Pr) taken as 0.71. The numerical experiments reveal that for Ra⩽8500, the flow always attains a steady state. In the steady regime, at very low Rayleigh numbers (Ra<300), it is shown that the velocity field is very weak and the heat transfer is predominantly by conduction. A series solution for the temperature field obtained by neglecting the fluid velocities is shown to agree well with the computed data for Ra<300. The convection takes place in the form of two cells with their interface aligned...

Two-dimensional buoyancy driven thermal mixing in a horizontally partitioned adiabatic enclosure

The dynamics of the transient, two-dimensional buoyancy driven thermal mixing of two fluid masses at different temperatures, initially at rest and confined to separate portions of a horizontally partitioned adiabatic enclosure, is investigated numerically within the framework of the Boussinesq approximation. The fluids are allowed to mix through a centrally located opening or vent in the partition. Apart from the geometric parameters, the dynamics is governed by the Rayleigh (Ra) and Prandtl (Pr) numbers. Spanning the range 500⩽Ra⩽104 at Pr=0.71 and unity aspect ratios of the vent and the enclosures, the dominant spatial and temporal flow structures, in the asymptotic approach of the system towards a state of thermomechanical equilibrium, have been identified. These dominant modes have been utilized to classify the flow dynamics observed at different Ra into three distinct flow regimes. An approach utilizing new scalar norms to quantify the instantaneous state of mixing and to track the mixing process in ...

A study of the asymmetric Malkus waterwheel: The biased Lorenz equations

In this work, the asymmetric case of the Malkus waterwheel is studied, where the water inflow to the system is biasing the system toward stable motion in one direction, like a Pelton wheel. The governing equations of this system, when expressed in Fourier space and decoupled to form a closed set, can be mapped into a four-dimensional space where they form a quasi-Lorenz system. This set of equations is analyzed in light of analogues of the Rayleigh Bernard convection and conclusions are drawn. The properties and behavior of the equations are studied and correlated to the physical model. Phase space behavior and linear stability analysis are used for this. Spectral analysis is used as a qualitative measure of chaos. Chaotic behavior is quantified through the calculation of the Lyapunov exponents and these are further correlated to the bifurcation diagrams for a conclusive analysis of the dynamical behavior of the system.

Extension of SMAC scheme for variable density flows under strong temperature gradient

An extension of SMAC scheme is proposed for variable density flows under low Mach number approximation. The algorithm is based on a predictor-corrector time integration scheme that employs a projection method for the momentum equation. A constant-coefficient Poisson equation is solved for the pressure following both the predictor and corrector steps to satisfy the continuity equation at each time step. Spatial discretization is performed on a collocated grid system that offers computational simplicity and straight forward extension to curvilinear coordinate systems. To avoid the pressure odd-even decoupling that is typically encountered in such grids, a flux interpolation technique is introduced for the equations governing variable density flows. An important characteristic of the proposed algorithm is that it can be applied to flows in both open and closed domains. Its robustness and accuracy are illustrated with a non-isothermal, turbulent channel flow at temperature ratio of 1.01 and 2.

A New Spatial Discretization Strategy of the Convective Flux Term for the Hyperbolic Conservation Laws

Abstract: In this work, a new spatial discretization scheme for flows governed by the hyperbolic conservation laws is proposed. The spatial discretization involves the concept of classical particle velocity upwinding (PVU) for the convective flux term in the hyperbolic conservation laws. The novelty of the approach lies in the use of the fluid particle velocity or the entropy wave speed at the cell interface to ascertain the upwind direction. The cell face convective fluxes are obtained from a first order or a second order upwind biased interpolation, depending on whether the cell under consideration lies in the vicinity of a discontinuity or in a region of steep gradients in the solution. The discontinuities or regions of steep gradients are detected by employing a smoothness indicator function as employed in some of the earlier studies. The proposed spatial discretization strategy has been combined with a two step, second order explicit time integration strategy for the application to the solution of the unsteady Euler/Navier-Stokes equations in the strong conservation form. Test cases involving two 1-D Riemann problems, three 2-D inviscid supersonic flow problems and a 2-D viscous supersonic flow problem, have been employed to establish the validity of the procedure and to assess the performance of the proposed strategy. The proposed PVU scheme performs quite favorably in comparison to conventional schemes. From the point of view of implementation, particularly in multidimensional scenarios, this strategy offers a good balance of accuracy and simplicity.