Kalita, Professor, Department of Mathematics, Indian Institute of Technology Guwahati, for the award of the degree of Doctor of Philosophy and this work has not been submitted elsewhere for a degree. Sailen Dutta, a student of the Department of Mechanical Engineering, Indian Institute of Technology Guwahati, for the award of the degree of Doctor of Philosophy has been carried out under our supervision and this work has not been submitted elsewhere for a degree.
INTRODUCTION
- Background
- Motivation
- Objectives
- The work
- Organization of the thesis
Over the years, researchers have developed HOC schemes for the ψ-ω form of the N-S equations and used them to successfully simulate fluid flow problems of varying degrees of complexity. Applying high-order compact schemes on non-uniform grids to the pure stream function form (or ψ-v form) of the Navier - Stokes (N-S) equations for heat and mass transfer problems of varying complexity.
COMPACT FINITE DIFFERENCE SCHEMES - A BRIEF DESCRIPTION
Introduction
In the case of compact/HOC schemes for the transient state equations, some schemes use 9 grid points at the nth time level and 5 grid points at the (n+ 1) time level. In this work, we have used a combination of two different schemes, except for the work detailed in Chapter 4, which uses a single(9,9)HOC scheme.
HOC scheme for convection-diffusion type equa- tions
Note that the compact scheme used for the biharmonic form of the N-S equations and the HOC scheme used for the temperature solution are all based on a 9-point template. The expressions for the coefficients Cij, Dij, Aij, Bij, Gij, Hij, Kij, Lij and Fij are detailed above.
Compact high order scheme for the biharmonic form of the N-S equation
Kumar and Kalita [94] first developed a high-order compact scheme on nonuniform grids for the stable version of equation (2.29), which, for a grid point (i, j) is given by. Thus, at the risk of sounding repetitive, the solution procedure for the algebraic system of equations is detailed in each chapter from now on.
NATURAL CONVECTION AROUND HEATED BODIES PLACED IN A SQUARE ENCLOSURE
Introduction
A reconstructed form of the scheme developed by Kumar and Kalita [94] for the biharmonic form of the steady-state N-S equations, and 2. For all three problems, we demonstrate the ability of the scheme to handle higher Rayleigh numbers.
Governing Equations - The bi-harmonic form
In all cases discussed in this chapter, the colder temperature Tc is taken as the reference temperature (T0 =Tc). Following the procedure described in section 2.3 of the previous chapter, we obtain the biharmonic form for the equations governing natural convection, which is given by .
Discretization and Numerical Procedure
Solution of algebraic systems
The computation of steady-state solutions of fluid flow problems governed by equations (3.9) and (3.10) involves an outer-inner iteration procedure. Preconditioning was particularly useful for high Rayleigh numbers where we have used the Lis library [105].
Validation of Code - Natural Convection in a dif- ferentially heated square cavity
- Grid Generation
- Results and Comparison
From Figures 3.3(a)-(f), one can clearly see the gradual thinning of the thermal boundary layers with increasing Raat of the hot and cold walls. As can be seen from the table, the errors fade with a convergence rate slightly higher than the theoretical rate of the scheme in [94].
Natural convection around a horizontal, heated, circular cylinder placed in an enclosure
- Grid Independence of the computed data
- Results and discussion
At δ = 0.2, for Ra= 105, the secondary vortices on the upper surface of the cylinder disappear as shown in Fig. When δ = 0.25, the space between the cylinder and the upper wall of the enclosure is too small. Thus there is a single plume down and two up on the upper surface of the cylinder.
Natural convection around a horizontal, heated, di- amond cylinder placed in an enclosure
- Results and Discussion
- A brief Nusselt Number analysis
- Comparison with the circular cylinder case
This property is the same for all positions of the cylinder and leads to a stable stratification in the core of the casing at high Rayleigh numbers. Taking this into account, we have calculated the flow rate for the special case of a round cylinder located in the center of the housing under the same boundary conditions that apply to the diamond cylinder. Even at the bottom of the objects, the convective heat transfer is much stronger for the circular cylinder, while at the top it is slightly lower than for the diamond cylinder.
Conclusions
The streamlined nature of the body favored the formation of boundary layers and its interaction with the fluid. Furthermore, the location of the cylinder was seen to have very little influence on the heat transfer pattern, as confirmed by the average Nusselt numbers. Finally, we have provided a brief comparison of the flow structure and heat transfer characteristics on a circular cylinder versus its diamond cylinder counterpart under the same conditions imposed on the boundaries.
DOUBLE DIFFUSIVE NATURAL CONVECTION IN A VERTICAL POROUS ANNULUS
Introduction
27] also numerically investigated DDNC in a square porous cavity, the bottom of which was partially heated and differentially salted. 8] numerically studied DDNC in a square porous housing when the bottom wall was partially heated and the right wall partially salted. In this chapter, an existing HOC flow function vorticity formulation on non-uniform grids is reconstructed for the simulation of DDNC in a vertical porous annulus.
The problem
Under these assumptions, the equations governing the conservation of mass, momentum, heat and concentration can be written in vector form as [2]. 4.6) where (ρC)f is the heat capacity of the liquid, and (ρC)p is the heat capacity of the saturated porous medium. A note on the assumptions: Since Darcy's law is assumed to be true, the slip boundary conditions prevail at the walls.
Discretization of the governing equations
- HOC Neumann Boundary Conditions
- Calculation of velocities
- Nusselt and Sherwood number
By performing some simple mathematical operations on the three equations above to eliminate second and third order derivatives, a third order accurate approximation of the Neumann boundary condition is obtained as Analogous to the ψ-ω form of the N-S equations ∇2ψ = −ω, it can be assumed for the current situation that ω. In most applications, the Nusselt (Nu) and Sherwood number (Sh) provide a summary of the overall HMT properties, respectively.
Grid Generation
The derivatives of T and S in (4.28) and (4.29), respectively, have been approximated using the formula derived in Section 4.3.1, and numerical integration for (4.30) and (4.31) has been performed using Simpsons 1.
Iterative solution procedure of the discrete equa- tions
Validation of algorithm and code
Results and discussion
- Influence of Aspect Ratio
- Influence of Radius Ratio
- Influence of Lewis number
- Influence of Buoyancy Ratio
- Influence of Thermal Rayleigh Number
However, the upper thermal eddy appears to be more chaotic in nature in the earlier stages of the flow (Fig 4.9(q)-(s)). The center of the solutal vortex also continues to move to the right end of the enclosure with an increase in κ. As Ra increases, the strength and size of the upper and lower vortices increase.
Conclusions
We proved that the strength of the current increases with Ra by tabulating the strength of the resulting vortices. We have also shown from Nusselt and Sherwood number plots that the convective heat and mass transfer rates increase with Ra not only for N =−2 but also for other values of N .
CONJUGATE HEAT TRANSFER IN SUDDENLY EXPANDING FLOW
Introduction
This flow behavior in a channel that expands suddenly with a large expansion ratio thus presents an opportunity to study the phenomenon of conjugate heat transfer. Next, we study the phenomenon of CHT in a symmetric channel that suddenly expands at low and large expansion ratios. Next, in Section 5.3 we simulate the CHT in a symmetric channel that suddenly expands with large expansion ratios.
Conjugate heat transfer in backward facing step flow
- Numerical scheme
- Grid generation
- Solution of system of equations
- Code validation
- Grid Independence
- Results and discussion
5.11 (b), asN increases with increasing Re, as well as from the temperature contour values in the liquid region (Fig. 5.9). 5.12(a) and 5.13(a)) the isotherms in the liquid region are loosely clustered close to the reattachment zone, whereas in the solid region they are linear along the plate. Predictably, the isotherms in the liquid region remain unaffected by the change in plate thickness (Fig. 5.18).
Conjugate Heat transfer in a suddenly expanding channel with large expansion ratios
- Grid generation
- Grid Independence
- Results and Discussion
This in turn shows that the rate of heat transfer will be maximum in the recirculation zone (to be further confirmed by the plots of interface temperature and local Nusselt number). The almost 50% difference in the heat transfer rate is only a reflection of the effect of expansion ratio. The thickness of the thermal boundary layer also decreases significantly, indicating an increase in the rate of convective heat transfer.
Conclusions
In general, in the liquid region, the isotherms were observed to converge near the reattachment point and spread transversely as they move downstream. In the solid slab, the temperature decreases in a vertical direction, with the lowest temperature value occurring near the reattachment point. The flow physics followed a similar pattern for both values of the Re's, with little variation in the isotherms and thermal layer thickness in the fluid beyond k = 10.
Introduction
The frequency associated with the periodic wake, the forces and moments acting on the body as well as the heat transfer parameters are a strong function of the shape and size of the body, the Reynolds number of the flow and the angle of incidence [129]. . This study investigated the effects of boundary conditions, domain flow extents, incidence angle, and β-blockage on the flow. Four main flow patterns were identified as functions of normalized corner radius (varying from 0–1) and incident angle (varying from 00–450).
The problem
This chapter is structured as follows: Section 6.2 presents the problem diagram, the preconditions and the associated equations. In section 6.5 we present the results for the simulation of forced convection over a diamond cylinder, and in section 6.6 we close this chapter. Another important non-dimensional parameter is the Strouhal number (St) which is given by. 6.5) where f is the frequency of vortex shedding.
Discretization and Numerical procedure
- Grid generation
- Solution of system of equations
The grid is generated so that the geometry of the diamond cylinder passes through the grid points. During this task, care must be taken to maintain the continuity of the gridlines in each direction of the overall computational domain. We now discuss the solution of the system of equations resulting from the discretization of the governing equations.
Code Validation - Forced convection over a circular cylinder at low Reynolds numbers
We have used bicongugate gradient stabilized (BiCGStab) method with preconditioning, where Incomplete LU decomposition is used as preconditioner. Point S is the separation point, φs is the separation angle, and Ls is the vortex length. The values of vortex length (Ls), separation angle (φs) and surface average Nusselt number (Nus) from the present calculation have been compared with well-established results i. a) Streamlines (left) and isotherms (right) for Re= 15 .
Results and Discussion
An enlarged view of these figures in the vicinity of the cylinder is given in figure 6.9 (a). The core of the vortex contains most of the heat and the heat is dissipated. We can observe that the hot fluid is trapped in the core of the shed vortices, as can be seen from the existence of local maxima of the contour values at the vortex centers.
Conclusions
