Nadeem Hasan

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

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...

Evolution to aperiodic penetrative convection in odd shaped rectangular enclosures

The purpose of this numerical study is to analyse the character of transition from laminar to chaotic convection in a fluid layer bounded by no‐slip walls in two space dimensions for varying aspect ratio odd‐shaped enclosures consisting of two rectangular chambers, with a linking horizontal enclosure. For a medium Prandtl number fluid (Pr=7), the numerical solution of two‐dimensional Navier‐Stokes momentum and energy equations with Bousinessq approximation has been carried out. It has been found that there are finite Rayleigh numbers Ra1, Ra2 and Ra3 for the onset of single, two and multiple frequency oscillatory motion at different spatial locations in the enclosure. As Ra is further increased period doubling is observed. The onset of strong chaos appears when Ra=Ra3. This system does not revert to steady state convection at high Ra as observed by other researchers for the case of Rayleigh‐Benard convection. Moreover, the period doubling transition process is consistent with the scenario of Ruelle, Takens and Newhouse. As Ra increases, the power spectrum, and time series of various dynamical variable signals, etc. all show an increasing degree of characteristics of chaos.

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 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.

On the Role of Coriolis Force in a Two-Dimensional Thermally Driven Flow in a Rotating Enclosure

In this work the role of Coriolis forces in the evolution of a two-dimensional thermally driven flow in a rotating enclosure of arbitrary geometry is discussed. Contrary to the claims made in some of the studies involving such class of flows that there is an active involvement of the these forces in the dynamics of the flow, it is shown that the Coriolis force does not play any role in the evolution of the velocity and temperature fields. This is theoretically demonstrated by recognizing the irrotational character of the Coriolis force in such class of flows. It is further shown that the presence of the irrotational Coriolis force affects only the pressure distribution in the rotating enclosure. The theoretical deductions apply quite generally to any geometry and thermal boundary conditions associated with the enclosure. The numerical results for the problem of two-dimensional thermally driven flow of air (Pr=0.71) in a circular rotating enclosure provide direct evidence of the theoretical deductions.

Vortex-shedding suppression in two-dimensional mixed convective flows past circular and square cylinders

Vortex-shedding suppression in two-dimensional mixed convective flows past circular and square cylinders is investigated numerically at two supercritical Reynolds numbers, Re = 60 and 100, at a fixed Prandtl (Pr) number of 0.71. The Richardson number (Ri) and free-stream orientation (α) are varied in the range [0, 1.6] and [0, π/2], respectively. The investigations involve the numerical solutions of mass, momentum, and energy equations subject to Boussinesq approximation in generalized curvilinear body-fitted coordinates. The critical Richardson numbers corresponding to the onset of suppression of vortex-shedding are determined for different free-stream orientations using the numerical data and the Stuart-Landau theory. For the case of circular cylinder, the critical Richardson number exhibits a “cosine-law” with respect to the free-stream orientation, while a non-monotonic trend is observed for the case of the square cylinder. By examining the near critical steady flow field data, two distinct components...

Peculiar Behavior of Thermally Stratified Channel Flow Under Non-Boussinesq Condition

Buoyancy affected turbulent flow in a channel is numerically simulated via LES, subjected to large temperature gradient. Temperature dependent fluid properties like viscosity (µ) and thermal conductivity (κ) considered as variable as Boussinesq assumption fails to capture the insight physics. With increased stratification, turbulent momentum and buoyancy fluxes decreased at an alarming rate in the core of the channel due to the formation of some wave like structures (IGW). The most striking feature produced by the temperature dependence of viscosity is flow relaminarization on the hot side of the channel (where viscosity is higher) and movement of IGW from core to hot side. Pronounced modifications in turbulent structure are observed qualitatively and quantitatively, and the deviation in the behavior is due to the non-Boussinesq assumption.