Computational Fluid Dynamics (CFD): Theoretical Fundamentals

In general, any engineering system or physical phenomenon can be studied in one of three ways:

1. Experimental approach
2. Analytical approach
3. Computational (numerical) approach

Take a simple case: air flowing through a converging tube, where we want to find the air speed at the outlet given its speed at the inlet. Any of the three approaches can be used. We can build a rig test with the control and measurement instruments needed to determine the exit velocity. We can solve it analytically by applying Bernoulli's equation between the inlet and outlet together with the continuity equation. Or we can build a numerical (computational) CFD model to find the result, as shown in the figure

The motion of any fluid can be simulated computationally by solving the flow governing equations, the Navier–Stokes equations. This is a set of equations that links all the variables of fluid motion, such as velocity, pressure, density and temperature. Because these equations are so complex, they are solved numerically rather than algebraically, and this numerical solution is what is done in Computational Fluid Dynamics (CFD)

The field of CFD is an interdisciplinary science at the intersection of three fields:

1. Fluid Mechanics
2. Mathematics
3. Computer Science

Since the simulation is carried out on a computer, and the only language a computer understands is the binary number system, made up of just 0 and 1, and since the governing equations are solved numerically rather than algebraically, the first step must be to convert the fluid domain under study from a continuous system, as it exists in reality, into a discrete system. This is known as discretization

The fluid's motion covers both space and time, so we have spatial discretization, which concerns the physical domain of the flow, and temporal discretization, which concerns its time domain. In space, the volume of the flow domain is divided into small units called cells. In time, the duration of the flow is divided into small units called time steps. The smaller the cells and the time steps, the more realistic the discrete system is, and the closer it comes to the continuous one. Within every cell, at every time step, the governing equations are solved numerically

So far, the results coming out of CFD software need to have their accuracy confirmed by comparing them with reference results from experiments. This is known as validation. The reason is that simulating turbulent flow is extremely complex and requires levels of simplification, so we need to know how much those simplifications affect the accuracy of the solution. We can only know that by comparing it with experimental test results

This raises a question: if we always need experimental results to confirm the accuracy of CFD models, what is the need for CFD in the first place? In my humble understanding, the answer has two points

First, validation does not have to be repeated for every case. It is enough to do it once, for one case, at the start, and then continue studying different cases with that validated CFD model. Second, numerical study beats experimental study in cost, since it is usually far cheaper than experimental testing, and in the level of detail in the results. For example, when simulating an air-conditioning system inside a room with one million cells and a time step of 0.1 seconds, you can know every property of the air at each of the million points, every 0.1 seconds. You can know the air's speed, temperature and humidity at a level of spatial and temporal detail that would be very difficult, or perhaps nearly impossible, to achieve with practical measuring instruments, or at best would be very expensive

These levels of simplification are numerous and vary from case to case. The first and most obvious one is the move from a continuous real-world system to a discrete one in space and time, as explained above. Beyond that:

Time: Time can be left out by using a steady-state solution, which assumes the fluid's properties do not change with time Geometry: A flow domain that is 3D in reality can be simplified to 2D, or to an intermediate 2.5D Symmetry: The domain can be reduced to a symmetric shape, halving the number of cells (with one symmetry plane) or quartering it (with two). For axisymmetric shapes, i.e. those symmetric about an axis of rotation, the case can be simplified to a segment at a certain angle that represents the whole shape. For example, a horizontal three-bladed wind turbine can be simplified to a single blade with a 120° segment Turbulence modeling: The most important simplification, and the one with the largest effect on the results, is how fluid turbulence is simulated. There are three general levels: RANS = Reynolds-Averaged Navier–Stokes LES = Large Eddy Simulation DNS = Direct Numerical Simulation

RANS is the most common in industrial applications and the most simplified approach to turbulence. DNS, as its name suggests, contains no simplification of turbulence at all. It is the most accurate and the most computationally expensive, and it is impractical for complex cases. LES sits between the two, but it needs relatively high computing power for complex cases compared with RANS

With the rapid progress in computing power and the emergence of what are called quantum computers, will we see, in the near future, CFD models with the least possible simplification, so that their results are reliable in their own right, without the need for validation against experimental results?