Computational Fluid Dynamics

Fluid flows (i.e., gas and liquid) are governed by partial differential equations (PDEs) which represent conservation laws for the mass, momentum, and energy. Computational fluid dynamics (CFD) is the art of replacing such PDE systems with a set of algebraic equations which can be solved numerically. As a rule, the algebraic systems to be solved are very large (millions of unknowns) but sparse i.e., most of the matrix coefficients are equal to zero.

In general, CFD aims to predict what will happen, quantitatively, when fluids flow, often with the complications of:

  • Simultaneous flow of heat.
  • Mass transfer (eg perspiration, dissolution).
  • Phase change (eg melting, freezing, boiling).
  • Chemical reaction (eg combustion, rusting).
  • Mechanical movement (eg of pistons, fans, rudders).
  • Stresses in and displacement of immersed or surrounding solids.

The fundamental basis of almost all CFD problems are the Navier–Stokes equations that describe the motion of viscous fluid substances. These balance equations arise from applying Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term.

The Navier–Stokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, given their wide range of practical uses, it has not yet been proven that in three dimensions solutions always exist, or that if they do exist, then they are smooth i.e., they do not contain any singularity. These are called the Navier–Stokes existence and smoothness problems. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a US$1,000,000 prize for a solution.

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