An Introduction to Computational Fluid Dynamics

Explore the Taylor series as an approximating tool that translates between the infinitesimal and the finite, enabling polynomial approximations of functions and their derivatives for CFD.

Explore Taylor series to approximate derivatives with multiple points, achieving higher order accuracy through forward, backward, and central schemes, for first and second derivatives, with automated coding.

Explore how Taylor series and finite differences enable numerical solutions of differential equations, from pendulum approximations to heat conduction, via mesh-based methods with Dirichlet boundaries.

Explore how an explicit finite-difference scheme updates the unsteady 1D heat equation, derive the update equation, and examine stability, boundary conditions, and parameter influence through Python code.

Compare explicit and implicit finite difference methods for the unsteady one-dimensional heat equation. Implicit schemes solve a tri-diagonal linear system, enabling larger stable time steps.

Explore how explicit and implicit schemes solve the 1d heat equation, and how von neumann stability and roundoff error via amplification factors and Fourier decomposition affect numerical convergence.

Analyze von Neumann stability of roundoff in explicit and implicit schemes for the 1d heat equation, deriving delta t <= delta x^2/(2 alpha) and noting unconditional stability of implicit methods.

Explore the one-dimensional uni-directional wave equation, its analytical solution, and numerical stability using a forward-in-time, backward-in-space scheme, including the CFL condition.

Analyze numerical errors in the wave equation using the ftbs scheme, derive the modified equation via Taylor expansions, and explain how CFL, stability, damping, and dispersion arise.

Explore macCormack's predictor-corrector scheme for the wave equation, revealing dispersive errors from odd derivatives, the role of CFL, and how artificial viscosity dampens wiggles near discontinuities.

Explore divergence and the substantial derivative within the Navier-Stokes framework, using a quasi one-dimensional converging-diverging nozzle to relate volume change to velocity gradients and prepare CFD solver design.

Explore how the divergence of velocity equals the fractional rate of change of volume in nozzle flows, using ux-driven expansion and the delta V concept for incompressible limits.

Explore divergence and the substantial material derivative in fluid dynamics, using a multivariate Taylor expansion and a thought experiment to illuminate how the Navier Stokes derivation arises from conservation laws.

Explore the conservation of mass in a moving fluid element, deriving the substantial-derivative equation for density and velocity divergence, and show that steady incompressible flow yields zero velocity divergence.

Apply Newton's second law to a moving fluid element, using the substantial derivative of momentum. Relate the rate of momentum change in each direction to body, pressure, and viscous forces.

Examine momentum conservation in x, y, z from Newton's second law for fluids, covering surface and body forces, and Newtonian fluids' linear relation between viscous stresses and strain rates.

Explore the conservation of energy in a fluid element, combining internal and kinetic energy under the first law, and the roles of work and energy transfer in computational fluid dynamics.

Explore the conservation of energy in fluids by distinguishing volumetric and surface heating, apply Fourier's law to heat conduction, and derive the one-dimensional heat equation.

Navigate the fidelity ladder to balance simplifying and complicating the Navier-Stokes equations, and learn nondimensionalization to reveal dimensionless numbers such as Reynolds, Prandtl, and Mach.

Learn nondimensionalization and key dimensionless numbers like Reynolds and Froude numbers to simplify the Navier–Stokes equations. Explore the fidelity ladder and two-dimensional incompressible Euler reductions for flow past a wing.

Investigate potential flow by deriving the Laplace equation from irrotational, incompressible two-dimensional Euler equations. See how this irrotational assumption yields a self-contained Laplace problem for CFD 101 practice.

Solve the Laplace equation for the potential phi under irrotational, incompressible flow, compute velocity as the gradient of phi, then obtain pressure via Bernoulli and report the pressure coefficient.

Demonstrates solving the Laplace equation for potential flow on a Cartesian mesh, updates the potential field, extracts velocity and pressure coefficients, and analyzes convergence and boundary effects around obstacles.

Explore the stream function–vorticity formulation for two-dimensional incompressible viscous flow, derive it from the Navier–Stokes equations, and apply it to the lid-driven cavity problem.

Explore the streamfunction–vorticity formulation for two-dimensional incompressible viscous flow, deriving omega and psi, applying nondimensionalization, and solving for lid-driven cavity boundaries.

Demonstrate the stream function-vorticity formulation for lid-driven cavity flow, updating omega and psi with finite differences under boundary conditions to reveal Reynolds-number–dependent vortices.

Derive the two-dimensional compressible Euler equations from the Navier-Stokes equations under inviscid assumptions and solve them with MacCormack's predictor-corrector scheme for isentropic vortex convection with periodic boundaries.

Explore the compressible Euler equations, shocks, and isentropic vortices, and learn how artificial viscosity stabilizes McCormack schemes in shock tube and related CFD simulations.