1.1.1.1. Computational Fluid Dynamics

1.1.1.1.1. Meaning of the Navier-Stokes Equations

\[ \begin{align}\begin{aligned}{\partial \vec V \over \partial t} + \vec V (\nabla \cdot \vec V) = -{\nabla p \over \rho} + \nu \nabla^2 \vec V\\\text{Unsteady term + Convective term = Pressure gradient term + Viscous term}\end{aligned}\end{align} \]

  • The momentum equation in the Navier Stokes Equations is a 2nd order, non-linear partial differential equation:

    • \(\vec V (\nabla \cdot \vec V)\) the convective term is the non-linear term

    • \(\nu \nabla^2 \vec V\) the viscous term is also the 2nd order term

1.1.1.1.1.1. Assumptions:

  • A Newtonian Fluid - shear stress proportional to strain

  • Incompressible flow - density constant

  • Isothermal flow - temperature constant

  • If the fluid is inviscid \(\nu = 0\). Then we have the Euler Equations. But it’s still non-linear:

\[{\partial \vec V \over \partial t} + \vec V (\nabla \cdot \vec V) = -{\nabla p \over \rho}\]

1.1.1.1.2. What is a solution?

  • The velocity field (vector)

  • The associated pressure field (scalar)

1.1.1.1.3. Why do we Need CFD?

  • There are very few known analytical solutions to the Navier Stokes Equations, e.g. when the convective term goes to zero, when there is flow in a pipe, flow between parallel plates and flow between concentric circles.

  • Systems may be difficult to test through experimentation - e.g. experiment doesn’t allow us to see inside or the instrumentation is limited, or it’s dangerous to do the experiment.

  • Faster and easier than experiment - CFD allows us to ask “what if” questions about a situation. Don’t have to build prototype.

  • CFD can be used in animation for films

1.1.1.1.4. Components of a CFD Model

1.1.1.1.4.1. Mathematical Model

  • A set of partial differential equations or integral-differential equations

  • Associated boundary conditions

  • The non linearity is a source of turbulence, vorticity, shock waves, combustion, multi-phase, bubble dynamics, evaporation, condensation. Therefore the model is associated with a target application - e.g. incompressible, inviscid, turbulent, 2D or 3D model

1.1.1.1.4.2. Discretization Method - the major part of CFD

  • Method for approximating the partial differential equations or the integral-differential equations by a system of algebraic equations, i.e. we have a PDE \(\mathcal{L} [u(\underline{x})] = f(\underline{x})\) and we need to convert it to arithmetic \(A\underline{x} = \underline{b}\)

  • The most important methods for doing this are:

    • Finite Difference (1950s)

    • Finite Element (1960s)

    • Spectral Methods (1970s)

    • Finite Volume (1980s)

    • Boundary Element Method

    • Particle Methods

  • Discretization has two aspects:

    • The Geometry (grid or mesh or particles) - gives us a vessel for the solution

    • The Model - all mathematical operators converted into arithmetic operations on grid

1.1.1.1.4.3. Analyse Numerical Scheme

  • All numerical schemes must satisfy certain conditions to be accepted:

    • Consistency

    • Stability

    • Convergence

    • Accuracy

1.1.1.1.4.4. Solve

  • Obtain grid/point values of all flow variables

  • Two situations:

    • Time dependent \(\Rightarrow\) ODEs

    • Steady \(\Rightarrow\) algebraic system of equations

  • To solve these equations we require:

    • Time integrators

    • Linear solvers

1.1.1.1.4.5. Post Processing

  • Visualization