I would also like to express my sincere thanks to all the faculty members of the Department of Mathematics for their help and cooperation. The effects of the Hartmann number and the Reynolds number on the flow behavior are discussed.
Background
The leading term in the truncation error determines the order of accuracy of the scheme. These types of schemes are desirable because a coarser grid can be used to get a better approximate result for the higher order accuracy of the scheme which reduces the computational cost.
Convective heat transfer
Therefore, it is important to understand the fluid flow and heat transfer characteristics of mixed convection in an ventilated cavity. Therefore, it is also important to understand the fluid flow and heat transfer characteristics of mixed convection in a lid-driven cavity.
Magnetohydrodynamics
The flow and heat transfer induced in the cavity is significantly different from the outer mixed convection boundary layer. The occurrence of mixed convection in cavities depends on the geometry and orientation of the cavity.
Objectives
The effect of electrical conductivity of the liquid and presence of magnetic field is: First, the movement of electrically conductive liquid across the magnetic lines of force generates currents according to Faraday's law of induction. Second, the combined magnetic field interacts with the induced current density to produce a Lorentz force on the conductor.
Motivation and thesis contribution
In the second part, the proposed HOC scheme is extended to a transitional one. Numerical results for transient fluid flow solutions in a lid-driven cavity with the same geometries as in the first part and their time variations are presented and discussed.
Organization of the work
This chapter contains the formulation of a high-order compact (HOC) scheme in Cartesian coordinate systems with non-uniform and non-orthogonal curvilinear lattices on complex geometries to solve two-dimensional (2D) second-order general partial differential equations with a non-homogeneous derivative source to solve. conditions. The schedule is fourth-order accurate in space and second-order accurate in time.
Introduction
The scheme is generalized in this thesis by adding more terms to the source function including the effects of heat transfer and magnetism, and the scheme is applied to various complex problems to study its strength and stability. A detailed discussion of the scheme in the transformed plane is presented in this chapter, and successful applications of this scheme to non-uniform curvilinear networks are also demonstrated.
Mathematical Formulations and Discretization ProcedureProcedure
Governing Equations
The governing equations to be studied in this thesis are second-order partial differential equations with mixed derivative and source terms involving derivatives. For a problem with complex geometries that have an irregular curvilinear mesh, the transformation is used to solve the problem on a uniform mesh.
Transformation of the equation
Discretization of the equation
To obtain a compact fourth-order spatial formulation for (2.4), each of the derivatives is compactly approximated by O(h4, k4). At the (i, j)th node, the fourth (fourth) order spatial approximation of equation (2.4) can be written as.
Solutions of algebraic systems
Numerical test cases
Comparison with analytic solutions
Hereφ, φC, φF respectively represent the exact solution, the solution on a coarse grid and the solution on a fine grid with twice the number of points in each direction. From table 2.1, it can be seen that the order of accuracy of the scheme is close to 4.
Lid-driven problem in wavy wall geometry
When Re = 400, the secondary vortices become larger and the primary vortex changes position slightly towards the center of the cavity. For M = 2 or 3, the same kind of flow properties are exhibited with the increase of the Reynolds number Re.
Time dependent form of the scheme on non uni- form gridform grid
Transformation of the equation
Discretization of the equation
Stability of the Scheme
Solutions of algebraic systems
Lid-driven problem in wavy wall geometry
In this case, one vortex initially appears near the upper right corner of the cavity. In the initial stage, there is one vortex near the upper moving wall and then two. In this case, there is initially only one vortex near the upper portion of the cavity, then a secondary vortex appears to the right of the primary vortex.
The Problem and the Governing Equations
The formulation of the eddy current function in the presence of a magnetic field with a porous medium can be written as When S = 0 and Ha 6 = 0, the equations are for problems without a porous medium but in the presence of a magnetic field. When S = 1 and Ha6 = 0, the equations are for problems with porous media in the presence of a magnetic field.
Numerical methods and Discretizations
Problem 1: Natural convection in presence of magnetic field without porous media:magnetic field without porous media
Results and discussions
The non-uniform heating increases the heat transfer rate in the center of the bottom wall compared to that of uniform heating. For different profiles of nonuniform heating at the bottom wall, the heat transfer rate distribution also changes accordingly. With the increase of P r from 0.71 to 7, the heat transfer rate at the bottom wall increases.
Problem 2: Natural convection in presence of magnetic field in cavity filled with fluid-saturatedmagnetic field in cavity filled with fluid-saturated
In this case the lower wall is heated, while the vertical side walls are adiabatic and the upper wall is cold. Near the bottom wall, the good temperature distribution (seen in the case of uniform heating) is affected and disturbed by the non-uniform temperature distribution at the bottom wall. The lower wall is heated (Th) uniformly while the vertical walls are adiabatic and the upper one is kept at temperature Tc.
Validation of the numerical results
Results and Discussions
It is clear from the streamline contours that with the increase of the Hartmann number, the strength of the flow decreases as the size of the streamlines decreases. It shows that with the increase of Darcy number Da, the heat transfer rate increases at the edges of the bottom wall, but towards the center, the heat transfer rate decreases with the increase of Da.
Conclusions
The analysis of flow and heat transfer in cover-driven cavities is one of the most widespread problems in the thermo-fluid field. They also derived a correlation for the average Nusselt number in terms of Prandtl number, Reynolds number and Richardson number. The inclined side walls are maintained at a lower temperature, and the bottom wall of the cavity is heated.
The Problem formulation and Governing equa- tions
They plotted the streamlines and isotherms for different values of Richardson number and also studied the variation of mean Nusselt number and maximum surface temperature at the heat source using Richardson number with different heat source lengths. The inclined side walls are kept adiabatic and the bottom wall of the cavity is kept at a uniform heat flow. Nusselt number (Nu) is the ratio of convective and conductive heat transfer perpendicular to the boundary.
Discretization method for the governing equa- tionstions
After discretization, the equation can be written in algebraic form, using the same procedure described in Chapter 2 for the time-independent case: To solve the algebraic system of equations (4.15), the biconjugate gradient stabilized method is used without preconditioning. Numerical results obtained are transformed back to the physical domain for discussion.
Results and discussion
- Validation of Results
- Effects of Richardson number, Ri
- Effects of Grashof number, Gr
- Effects of Prandtl number, P r
- Effects of bottom width with fixed vertical length of the trapezoidal cavitytrapezoidal cavity
- Effects of vertical height of the trapezoidal cavity
- Heat transfer rate at the bottom boundary
At Gr = 1000 it is seen that a minor cell is formed in the lower left corner of the cavity together with the primary cell. Also from the isotherm contours it can be seen that the heat transfer rate increases with the increase in width of the lower wall. With the increase of Grashof number, the local Nusselt number decreases near the center of the lower wall.
![Table 4.1: Comparison of the location(X c , Y c ) of the primary eddy with the stream function value ψ c between the present scheme and [54]](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10531232.0/91.918.145.804.163.744/table-comparison-location-primary-stream-function-present-scheme.webp)
Conclusions
An analytical solution of the mixed convection heat transfer in high cavities is presented by Arpaci and Larsen [3]. In this chapter, the mixed convection is studied in a rectangular enclosure where the bottom wall of the enclosure is considered to be wavy. Openings of the inlet and the outlet are kept as 1/4th time of the wall height of the enclosure.
Problem and the governing equations
The inlet is located at the lower part of the vertical wall, where the outlet is fixed at the top of the opposite vertical wall of the enclosure. A colder fluid enters the enclosure through the inlet and a constant heat flux is applied to the portion of the wall above the inlet at the left vertical wall. We have the stream function ψ = 0 on the bottom wall and part of the right wall below the outlet.
Discretization method for the governing equa- tionstions
To measure the heat transfer at a boundary within a fluid, the Nusselt number (Nu), which is the ratio of convective to conductive heat transfer across (normal to) the boundary, is calculated. The heat transfer coefficient in terms of local Nusselt number (Nu) is defined by [27], Nu= 1. After discretization, using the same procedure followed in Chapter 2 for time-independent case, the equation can be written in algebraic form. 5.15).
Comparison of Results
To solve the algebraic system of equations (5.15), the biconjugate gradient stabilized method is used without preconditioning. The chosen convergence criterion in this task is considered to be that the error above is bounded by 0.5×10−4. It is noted that the results obtained using the proposed scheme are in good agreement with the results discussed in [82].
Results and Discussion
There is no significant change in the flow structure in the upper half of the cavity due to multiple waves at the lower wall, but vortices are formed at each indentation of the wavy wall. Two separate vortices are formed in the center of the cavity near the hot wall. Heat transfer near the inlet is maximum while it decreases as it moves away from the inlet towards the top of the cavity.
Conclusions
The general trend of Figures 5.25-5.27 is that the local Nusselt number decreases along the vertical height. The local heat transfer increases with the increase of Re from 10 to 100, but when Re increases to 200, the heat transfer rate is lower than that for Re= 100. This shows that with the increase of Ra(i.e. the increase in heat flux q at the left wall) the local heat transfer rate increases.
Dilated channel problem
Geometry of the problem
In the case of lateral dilated canal, the expansion occurs only on one side (the upper wall), while for axisymmetric case, the expansion is on both sides and symmetrical about the center line i.e. the expansions are on both the upper and the lower walls as shown in the Figs 6.1 and 6.2. For lateral widened channel geometry, bottom wall is taken as the x-axis and for the other case, centerline is considered as the x-axis.
Governing equations for dilated channel problem
Boundary conditions
If we introduce flow function (ψ) and vorticity (ζ) as:. the above equations can be written as. In the absence of any magnetic field, the governing Navier-Stokes equations in terms of current function and vorticity become: and for axially symmetric dilated channel:. 6.16) Initially the entire system was at rest and at the inlet a velocity profile is imposed as discussed above.
Numerical methods for solving the governing equationsequations
Grid independency
In general, any desired accuracy of the solution is achieved by reducing the mesh size in the calculation. Reducing the grid size means increasing the number of grid points, which in turn affects the computational cost. To solve this situation, we need a standard mesh size so that further reduction of the mesh size does not affect the results of the problem.
Results and Discussions
By increasing the Hartmann number to Ha= 15, this vortex disappears and there is no reverse flow in the extended channel. Increasing Re to 500 increases the number of vortices on both sides of the widened channel. As the length of the dilated part increases, the speed of the flow increases, and the number of vortices also increases.
Conclusions
However, the local Nusselt number reaches its maximum value at the corner points of the lower wall and its minimum at the center. For non-uniform heating of the bottom wall, the heat transfer rate is 0 at the corners of this wall. A sinusoidal type of local heat transfer rate is observed showing minimum at the corner points as well as at the center of the bottom wall.
Future scopes
Belkadi, Laminar natural convection in an inclined cavity with a corrugated wall, International Journal of Heat and Mass Transfer pp. Moreau, Buoyancy driven convection in a rectangular enclosure with a transverse magnetic field, International Journal of Heat and Mass Transfer pp. Kuwahara, Mixed convection in a driving cavity with a steady vertical temperature gradient, International Journal of Heat and Mass Transfer p.
