Next, an HOC treatment is developed for the streamfunction-vorticity (ψ-ω) formulation of the two-dimensional unsteady, incompressible, viscous Navier-Stokes equations on polar grids, specifically designed for motion past circular cylinder problems. 47 4.7 For Re= 60, (a) streamline and (b) vorticity contours (corresponding to the . peak value of the lift coefficient) for the time-periodic solution.
Background
Another way to obtain higher order compactness is by using the original differential equation to replace the truncation principal error (TE) terms of the standard central difference approximation. Carey [106] in the formulation of the steady-state stream function vorticity (ψ-ω) of the N-S equations and by Kalita et al.
Motivation
Very few attempts have been made to develop the HOC scheme on non-uniform grids for the convection and diffusion equations using the conventional technique of transformation from the physical plane to the computational plane. 62, 64] extended this work to the convection-diffusion equations as well as the vorticity formulation of the Navier-Stokes equations for the transition function of the stream on non-uniform grids in the Cartesian coordinate system.
Objectives
The Work
For the higher range of Re, we calculate the solution in the initial stages of the flow. We also compare the simulations of the flow patterns calculated by us with the experimental flow visualizations and numerical results available in the literature.
Organization of the work
In equation (2.1), the magnitude of the convection coefficients determines the ratio between convection and diffusion and is sometimes referred to as the Reynolds number (Re). It may be mentioned that the majority of the previous efforts to develop HOC schemes on cylindrical polar coordinates were limited to the Poisson equation on uniform grids.
Mathematical Formulations and Discretization Procedures
Thus, the extension of the current scheme from polar coordinates to a general rectangular curvilinear coordinate system in 2D is a mere formality. In such cases, we can resolve the 0/0 form using L'Hospital's rule whenever possible or use a local Cartesian grid at the singularity point [9].
Solution of algebraic systems
For the inner iterations, the calculations were stopped when the norm of the residual vector¯r=F −AΦ arising from equation (2.16) fell below 0.5×10−6. For the steady-state solution of the problems governed by the N-S equations, it was assumed that steady state was reached when the maximum ω error between two consecutive outer iteration steps was smaller than 1.0×10−9.
Numerical experiments
Test problem 1: a problem of pure diffusion
It should be noted that for coupled nonlinear PDEs (such as the ψ-ω form of the N-S equations) an iterative solution procedure must be used. The latter iterations can be called inner iterations, which must be performed on every outer iteration with updated data.
Test problem 2: Flow past an impulsively started circular cylinder: 15
We also compare the vorticities along the surface of the cylinder for the Reynolds number range considered here with those of references [38, 44] in Figure 2.8. In Figure 2.18(a) we present the uandv velocities along the radial line θ= 0, obtained by our calculation on an 81×81 grid for Re = 55, together with the experimental results of [48].
Conclusion
In the first problem Dirichlet boundary conditions are used and for the other two have compact higher order approximations. The robustness of the scheme is illustrated by its applicability to the last two fluid flow problems of different physical complexities and their accurate calculations.
The governing equations
Mathematical Formulation and Discretization Procedure
To achieve this, the original equation (3.1) is treated as an auxiliary relation that can be differentiated to obtain higher order derivatives. After all approximations are substituted for the derivatives, the spatial HOC approximation of equation (3.1) can be written as.
Conclusion
Thus, (3.10) is the HOC approximation of the vorticity equation (3.1), which is at least third-order accurate in space and second-order accurate in time.
Introduction
We now have enough experimental data that can be compared with the outcome of the numerical results, which paves the way for the calculation of complicated and extended flow phenomena for Reynolds numbers that have so far not been investigated by experimental investigations. In each case, our solution agrees very well, both qualitatively and quantitatively, with established numerical and experimental results, confirming the effectiveness of the proposed scheme.
The problem and the governing equations
However, the robustness of the scheme is better realized when it captures the periodic nature of the flow for Re = 60 and 200 characterized by vortex shedding represented by the von K´arm´an street and also by the fact that it very accurately captures the so-called secondary phenomena for moderate Re, and α and β phenomena for higher Re. The boundary conditions forψ on the surface of the cylinder can be derived from the velocities in (4.8) as.
Approximation of the Boundary Conditions
Calculation of Drag and Lift coefficients
The Grid used
Solution of algebraic systems
It can be noted that for the coupled nonlinear PDEs (such as the ψ-ω form of the N-S equations) an iterative solution procedure must be applied to solve the matrix equation of type (4.21) at each time step. For the inner iterations, the calculations were stopped when the norm of the residual vector¯r = B −AΦ (φ being ω or ψ) resulting from equation (4.21) fell below 0.5×10−6.
Results and Discussion
As mentioned earlier, for the flow past an impulsively started circular cylinder, steady-state is possible up to Re= 40. From these tables it is clear that a grid of size 101×101 and a far field given by R∞= 75 is enough for accurate resolution of the flow.
Flows for Re = 60 and 200
In figure 4.12(a) we compare the radial velocity values obtained by our calculations at earlier stages along the flow axis with those of the experimental results of Bouard and Coutanceau ([26]). In figure 4.13(b) vorticity distribution along the solid surface is shown for the same time interval.
Flows for 300 and 550
Comparison of the streamlines captured by our scheme at t= 2.5 with the experimental result of reference [26] is shown in figure 4.23. In Figure 4.24 (b), we show the corresponding vorticity distribution along the solid surface for the same Reynolds number.
Conclusion
The study of the flow past a rotating cylinder was started in the middle of the first half of the last century by Prandtl and Tietjens. We also compare the simulations of the flow patterns we have computed with experimental flow visualizations and numerical results available in.
The Governing Equations
In all cases, our numerical results are in excellent agreement with existing results. In Section 5.4, we provide detailed discussions of our numerical results and comparisons with existing experimental and numerical results.
Approximation of Boundary Conditions
NOTE: Three parameters used in the calculation are of utmost importance to obtain an accurate numerical solution of this flow. From a computational point of view, this third parameter plays a crucial role in the accurate time matching of the numerical results with the existing experimental ones.
Results and Discussion
In figure 5.6(b) we show the vorticity values on the cylinder surface at different time levels. In figure 5.12(b) we show the vorticity values on the cylinder surface at different time levels.
At the same time, a small tertiary vortex appears between the secondary vortex and the surface of the cylinder below the x-axis and persists until t = 6 (figure 5.23(e)) approximately. Simultaneously, the secondary vortex detaches from the cylinder surface and moves downstream.
In Figure 5.54(a) we show the time variation of the calculated centers of the first vortex with that of the experimental results of Badr et al. Finally, in Figure 5.54(b) we show the time variation of the calculated centers for the first and second vortex for α = 3.
Conclusion
They further provided a brief analysis of the one-dimensional immersed interface method for steady-state and time-dependent problems. We also perform some convergence and corresponding analyzes for the approximation of the steady-state case.
Mathematical Formulations and Discretization Procedures
Steady state case
Our scheme is at least third-order accurate on the regular points and exactly third-order accurate on the irregular points. We use different approaches to discretize the given differential equation at the regular and irregular points.
Unsteady case
With this we have exactly (n−1) equations in the same number of unknowns and φi's can be calculated at all the specified points. As in the steady-state case, using one-sided difference at the points xj and xj+1, equation (6.27) takes the form.
Convergence and Related Analysis
Lemma-4: If the coefficients of the given differential equation are variables of x and smooth (up to the minimum second order), Lemma-3 can be written as follows with the same order. But this difference equation is an accurate third-order approximation of the real differential equation (6.66) at every lattice point from which it immediately follows.
Numerical Examples
Problem 1
Here, σ∗ and σh respectively denote the calculated flux values using our and reference schemes [79]. These observations are consistent with the development of a scheme where the condition d1, d2 → 0 is a prerequisite for exact solutions of our scheme.
Problem 2
This is also observed in table 6.3 where it is seen that our calculated solutions are almost exact at all the grid points. It is also encouraging to note that both those schemes in reference [7] used 41 grid points while our scheme produced almost exact results using only 10 points.
Conclusion
The next chapter discusses the extension of the formulation to two-dimensional cases. In this chapter we borrow some of the ideas from Chapter 6 to propose a new approach for numerically solving 2D elliptic equations with discontinuous coefficients and single source terms on uniform space grids. βφx)x+ (βφy)y +κ(x, y)φ =f(x, y), (x, y)∈Ω, (x∗, y∗)∈Γ (7.1) with some specified boundary conditions, where β , κ and f are piecewise continuous, but can have a finite jump discontinuity over some interface (a curve in 2D case).
Mathematical Formulation and Discretization Procedure
Irregular points along x-direction (at a fixed y-level)
In this case, the irregular points on a particular y-line (ie, for a fixed y-value) are the points directly adjacent to the interface (located either to the left or to the right) and the intersections between the interface and that y-line will be the interface points for that y-line. The value of that grid point exactly on the interface will then be calculated by interpolation.
Irregular points along y-direction (at a fixed x-level)
Numerical Examples
It is worth noting that our computational solutions with a grid as coarse as 23 × 23 are relatively close to the analytical ones.
Conclusion
Unlike previous HOC schemes which were only able to handle unit diffusion coefficients in Cartesian coordinates, this new class of HOC schemes is able to handle variable second-order derivative coefficients that appear in the type of convection equations -diffusion: both in Cartesian and other curvilinear orthogonal coordinate systems. The second part of the PhD work is about grouping some existing HOC methodologies with new discretization strategies at certain points of the physical field in order to capture discontinuities and solve circular interface problems in Cartesian coordinates.
Achievements
Scope for Future Works
