The computational domain is bounded by edges 2-D or faces 3-D on which boundary conditions are applied. The numerical equations are then solved in each cell of the mesh. A mesh is also called a grid. The transport equations of fluid mechanics are conservation equations.

For example, the continuity equation is a differential equation representing the transport of mass, and also conservation of mass. The Navier-Stokes equation is a differential equation representing the transport of linear momentum, and also conservation of linear momentum. In other words, the equations cannot be solved alone, but must be solved simultaneously with each other. This is the case with fluid mechanics since each component of velocity, for example, appears in the continuity equation and in all three components of the Navier-Stokes equation.

Analysis Nodes are points along an edge of a computational domain that represent the vertices of cells. In other words, they are the points where corners of the cells meet. Intervals, on the other hand, are short line segments between nodes.

Intervals represent the small edges of cells themselves. In Fig. P there are 6 nodes and 5 intervals on the top and bottom edges. There are 5 nodes and 4 intervals on the left and right edges. Discussion We can extend the node and interval concept to three dimensions. Chapter 15 Computational Fluid Dynamics C Solution For a given computational domain with specified nodes and intervals we are to compare a structured grid and an unstructured grid and discuss.

Analysis We construct the two grids in the figure: a structured, and b unstructured. Discussion Depending on how individual students construct their unstructured grid, the shape, size, and number of cells may differ considerably. Analysis We construct the two grids in the figure: a structured, and b unstructured polyhedral. We show two other options in c and d.

There are many possible answers for the polyhedral mesh, depending on how large you want your cells to be.

### Introduction to CFD Analysis: Theory and Applications

There are 22 cells in polyhedral grid b. There are some cells with 3 sides, 4 sides, and 5 sides, as required.

Compared to the triangular mesh with 36 cells, we have reduced the cell count considerably. In c and d , there are 21 cells and 18 cells respectively. In case d we have reduced the cell count below that of even the structured grid. In that case, 3 of the cells have 6 sides each. The cell reduction is particularly useful in large 3-D problems where CPU time and computer memory are important limitations.

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Discussion Note that the node distribution along the boundaries is identical in each case, but we have great flexibility in how we create the grid. Depending on how individual students construct their unstructured grid, the shape, size, and number of cells may differ considerably.

## Introduction to CFD Analysis with Practical Examples | SimScale Blog

Analysis We list the steps in the order presented in this chapter: 1. Specify a computational domain and generate a grid.

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Specify boundary conditions on all edges or faces. Specify the type of fluid and its properties. Specify numerical parameters and solution algorithms. Apply initial conditions as a starting point for the iteration. Iterate towards a solution. After convergence, analyze the results post processing. Calculate global and integral properties as needed. Discussion The order of some of the steps is interchangeable, particularly Steps 2 through 5.

Analysis Flow separates over bluff bodies, generating a wake with reverse flow and eddies downstream of the body. There are no such problems upstream. Hence it is always wise to extend the downstream portion of the domain as far as necessary to avoid reverse flow problems at the outlet boundary.

Discussion The same problems arise at the outlet of ducts and pipes — sometimes we need to extend the duct to avoid reverse flow at the outlet boundary. Analysis a In a CFD solution, we typically iterate towards a solution. In order to get started, we make some initial conditions for all the variables unknowns in the problem.

These initial conditions are wrong, of course, but they are necessary as a starting point. Then we begin the iteration process, eventually obtaining the solution. We construct a residual by putting all the terms of a transport equation on one side, so that the terms all add to zero if the solution is correct. As we iterate, the terms will not add up to zero, and the remainder is called the residual.

As the CFD solution iterates further, the residual should hopefully decrease. As the iteration proceeds, the variables converge to their final solution as the residuals decrease. Examples include plotting velocity and pressure fields, calculating global properties, generating other flow quantities like vorticity, etc. Post processing is performed after the CFD solution has been found, and does not change the results. Post processing is generally not as CPU intensive as the iterative process itself. Discussion We have assumed steady flow in the above discussions.

Analysis a With multigridding, solutions of the equations of motion are obtained on a coarse grid first, followed by successively finer grids. This speeds up convergence because the gross features of the flow are quickly established on the coarse grid, and then the iteration process on the finer grid requires less time. Then, an artificial time is used to march the solution in time. In some cases, this technique yields faster convergence. Analysis We may apply the following boundary conditions: outflow, pressure inlet, pressure outlet, symmetry to be discussed , velocity inlet, and wall.

The curved edge cannot be an axis because an axis must be a straight line. The edge cannot be a fan or interior because such edges cannot be at the outer boundary of a computational domain. The symmetry boundary condition merits further discussion. Numerically, gradients of flow variables in the direction normal to a symmetry boundary condition are set to zero, and there is no mathematical reason why the curved right edge of the present computational domain cannot be set as symmetry. However, you would be hard pressed to think of a physical situation in which a curved edge like that of Fig.

P would be a valid symmetry boundary condition.

## Essentials of Computational Fluid Dynamics

Discussion Just because you can set a boundary condition and generate a CFD result does not guarantee that the result is physically meaningful. Analysis The standard method to test for adequate grid resolution is to increase the resolution by a factor of 2 in all directions if feasible and repeat the simulation. If the results do not change appreciably, the original grid is deemed adequate. If, on the other hand, there are significant differences between the two solutions, the original grid is likely of inadequate resolution.

In such a case, an even finer grid should be tried until the grid is adequately resolved. Discussion Keep in mind that if the boundary conditions are not specified properly, or if the chosen turbulence model is not appropriate for the flow being simulated by CFD, no amount of grid refinement is going to make the solution more physically correct. Chapter 15 Computational Fluid Dynamics C Solution We are to discuss the difference between a pressure inlet boundary condition and a velocity inlet boundary condition, and we are to explain why both pressure and velocity cannot be specified on the same boundary.

## Introduction to Computational Fluid Dynamics (CFD)

Analysis At a pressure inlet we specify the pressure but not the velocity. At a velocity inlet we specify the opposite — velocity but not pressure. To specify both pressure and velocity would lead to mathematical over- specification, since pressure and velocity are coupled in the equations of motion.

When pressure is specified at a pressure inlet or outlet , the CFD code automatically adjusts the velocity at that boundary. In a similar manner, when velocity is specified at a velocity inlet, the CFD code adjusts the pressure at that boundary. Discussion Since pressure and velocity are coupled, specification of both at a boundary would lead to inconsistencies in the equations of motion at that boundary.

Analysis The inlet is a velocity inlet. The outlet is a pressure outlet. All other edges that define the outer limits of the computational domain are walls. Finally, there are three edges that must be specified as interior. These are all labeled in the figure below. Otherwise the CFD solution will not be correct. Analysis Since the fan helps to push air through the channel, the inlet pressure will adjust itself so that less inlet pressure is required.