What is the role of numerical linear algebra in computational fluid dynamics?

It's like the role that teeth play in our digestive system. It crunches the number solving the system of eqns. and answers a PDE. Un papel clave!

In engineering problems, we often come across linear and non-linear differential equations. These governing equations must be solved, and the analytical method of solving them can not be used due to their complexity. However, it is possible to use computational fluid dynamic techniques to obtain computer-based solutions for these complex equations. Different discretization methods, such as the finite element method, finite volume method, and finite difference method, can be used in the CFD modeling of these engineering problems.

Many discretization methods (on governing equations; the PDEs & ODEs) would led to a set of linear algebraic equations! So they play an important role in CFD and knowing them and to how to solve them is a key in CFD solvers!

Numerical linear algebra in CFD simulations enables us to study complex fluid flow phenomena with much more accuracy and also efficiency by providing efficient and precise algorithms for solving large and complex systems of linear equations that appear in CFD simulations. For instance, FSI simulations involve the interaction between fluids and solid objects, such as the flow of air over an aircraft wing or the flow of blood through blood vessels. These simulations require solving coupled systems of equations that describe the motion of both the fluid and the solid object. involve partitioning the computational domain into smaller sub-domains and solving the sub-problems iteratively.Cases which can't be solved by analytical methods.

Computational Fluid Dynamics (CFD) is studying fluid flow using computational methods such as finite volume method and finite difference method. The governing equation for fluid flow is called Navier-Stokes equation which is developed based on conservation of mass and momentum of a fluid element. This equation has no analytical solution (except for very few cases) due to it non-linearity. Deriving and manipulating the equation itself requires solid understanding of linear algebra and calculus. Moreover, solving the equation is done numerically, commonly using computer algorithms, which require solid background in numerical methods.

That's actually true. Since the CFD governing equations are in the partial derivative form with exact final solution not known except the initial conditions enables it to solve iteratively using those conditions. Discretization methods are used to chop a continuous function (i.e., the real solution to a system of differential equations in CFD) into a discrete function, where the solution values are defined at each point in space and time. Discretization simply refers to the spacing between each point in your solution space.

Numerical linear algebra may provide algorithms and may guide in getting tools for solving the systems of linear equations from discretization of fluid flow problems. CFD has numerical solution of PDEs that govern fluid flow, and these PDEs often lead to large systems of linear equations.

One more thing to add, while there are a set of commercial and open-source CFD tools in which engineers can use to simulate their systems, without the need to code, I highly recommend having a solid understanding of CFD concepts and methods, with high focus on the mathematical aspect (linear algebra and calculus) of this science. This enables the CFD engineer to model and analyze his/her system more effectively and accurately.