**4. Summary of other important CFD advances**

Here we will discuss some technical innovations that have allowed CFD to become a widely available tool for researchers.

#### **4.1 Coupling of velocity–pressure equations**

Regardless of whether a simulation is steady or unsteady, an iterative process must be undertaken wherein solutions are fed back into the coupled equations and then the solutions are updated to improve accuracy. The pressure and velocity fields are coupled via the governing equations and it is challenging to calculate them simultaneously. A series of approaches was developed that began with the so-called SIMPLE algorithm (Semi-Implicit Method for Pressure Linked Equations) [4, 5]. These references provide extensive detail on the algorithm, which has been joined by a modified SIMPLER version, the SIMPLEC method, and the PISO method. All of these methods continue to be used in today's algorithms. Use of these approaches greatly improves the stability of the pressure–velocity coupling.

#### **4.2 Multigrid calculations**

Another major innovation has been the utilization of multigrid simulations to increase the speed of solution. With a multi-grid method, the governing equations are first solved on a coarse mesh. This solution is then sent to a new mesh that contains smaller elements and iterations with the new mesh are performed. The solution is again passed to a further refined mesh and the process continues until the mesh is identical with the cells that the user has defined (for example in **Figure 5**). Following a converged *Introductory Chapter: A Brief History of and Introduction to Computational Fluid Dynamics DOI: http://dx.doi.org/10.5772/intechopen.97235*

solution on the smallest cells, the process is reversed, information is passed to increasingly coarse meshes until the algorithm arrives back at the initial, coarse mesh.

Through this process, solution information is transferred from the boundary conditions to the interior of the solution domain much faster than if the solution were obtained only on the finest computational mesh. It should be recognized that information is able to travel the distance of a cell size in a single iteration. If the solution domain is composed entirely of small elements, it means many iterations must be processed simply to transfer boundary information to the interior of the domain. The multi-grid approach largely solves this problem.

#### **4.3 Parallel computing**

Parallel computing refers to the simultaneous use of multiple processors during the iterative procedure. The software separates the computational mesh and parses different parts of the mesh to individual CPUs. Each CPU performs iterations on its batch of cells. The results are then collected, information is sent between the batches of cells, and the process is repeated.

To provide an example, a simulation that requires 10,000,000 elements that is solved with 10 processors would involve an allocation of approximately 1,000,000 elements to each processor. Clearly solving for 1,000,000 elements is far faster than the entire solution, and thus a time savings is achieved. On the other hand, there is a loss of efficiency in the partitioning (separating the elements into individual batches for each processor and passing information between the processors).

Parallel processing speed is measured as follows:

$$\text{Speed Up Time} = \frac{\text{Time Required for Simple Process}}{\text{Time Required for Multiple Process}} \tag{1}$$

**Figure 6.** *Example of elements that follow curved boundaries.*

Even with standard personal computers which typically have 4–8 CPU cores, a significant speed up time can be achieved. However, parallel computing is not advised for problems with a small number of elements. As a rule of thumb, each processor core should have approximately 100,000 computational elements assigned to it in order for there to be a time savings. For problems with 100,000 elements, a single processor is typically more efficient. For problems with more than approximately 200,000 elements, two processors are recommended, and so forth.

#### **4.4 Complex element shapes**

The last technical innovation that will be discussed is the advancement in element shapes. Initially, computational elements were made with simple shapes (squares and rectangles). While rectangular elements are suitable for rectangular solution domains, they are not necessarily appropriate for more complex shapes. For example, at curved boundaries, the elements would follow the boundary using a stair-step deployment. Modern day simulation algorithms generally no longer require rectangular elements. The governing equations can be solved with arbitrarily shaped elements. As a representative example, we provide a mesh that was recently used to simulate water flow past an oceanography instrument called the eXpendable BathyThermograph (XBT) As evident from the figure, a mesh has been deployed that is able to exactly follow the curved surface of the device and accurately depict the flow region (**Figure 6**).
