Computational Fluid Dynamics - Session 02 Notes

Basic Definitions of Differential Equations

  • Ordinary Differential Equations (ODEs)
  • Partial Differential Equations (PDEs)

Fluid and Thermal Problem Types

  • Equilibrium Problems: Steady-state problems.
    • Examples:
      • Steady-state temperature distribution in a solid rod.
      • Equilibrium stress distribution of a loaded structure.
      • Steady fluid flows.
    • Governed by elliptic PDEs.
  • Marching Problems: Transient problems.
    • Include: transient heat transfer, all unsteady flows, and wave phenomena.
    • Governed by parabolic or hyperbolic PDEs.

Definitions of Differential Equations

  • First Derivative: Definition and geometric interpretation of the 1st derivative of a function f(x) at point x = x_0.
  • Second Derivative: Definition and geometric interpretation of the 2nd derivative of a function f(x) at point x = x_0.
  • Ordinary Differential Equation (ODE): Definition and examples of ODEs encountered previously.
  • Partial Derivative: Definition of the partial derivative of f(x, y) with respect to x at (x, y) = (x0, y0).
  • Laplace Equation: Example of a PDE.
  • Partial Differential Equation (PDE): Definition of a PDE.

Elliptic PDEs

  • Model Elliptic PDE: Laplace equation.
    \nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0 …(1)
  • Complete solution in a rectangular subdomain requires providing data on all domain boundaries, such as the value of the function \phi on all four boundaries.
  • Solutions are found by imposing conditions on the unknown function on ALL boundaries, known as boundary-value problems.
  • Example: 1D steady heat transfer in a rod of length L. Find \phi(x) = T(x).
  • A perturbation at one point affects the solution of the entire domain.
  • Describe steady-state conductive heat transfer and inviscid, incompressible, irrotational flow.

Parabolic PDEs

  • Describe time-dependent problems involving significant diffusion: unsteady viscous flows, unsteady heat conduction.
  • Prototype Parabolic PDE: Diffusion equation.
    \frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} …(2)
  • Sample Problem: Unsteady heat transfer in a rod of length L. Determine T(x, t).
  • Requires boundary conditions and an initial condition.
  • A disturbance at a point x1 in the solution region interior (0 < x1 < L) and time t1 can only influence events at later times (t > t1).
  • Solutions are determined by imposing boundary conditions on all domain boundaries and initial conditions on the interior domain; these are initial-boundary-value problems (IBVPs).

Hyperbolic PDEs

  • Appear in time-dependent processes with negligible energy dissipation: small amplitude vibrations, sound propagation.
  • Model Hyperbolic PDE: Wave equation (where c is the wave speed).
    \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} …(3)
  • Hyperbolic PDEs also yield initial-boundary-value problems.
  • Key to solving and studying shock discontinuities in transonic and supersonic flows (aircraft and missile aerodynamics, gas turbines, ramjets, and scramjets).

Domain of Dependence

  • Domain of Dependence: The region of the (x, t) plane on which the solution at any given point (x_0, t^*) depends.
  • Parabolic PDE: Solution at point (x^, t^) depends on data at t < t^*.
  • Elliptic PDE: Solution at point (x^, t^) depends on data at all points.

Summary

  • Equilibrium Problems: Steady physical processes, governed by elliptic PDEs, solution depends on boundary data only.
  • Marching Problems: Transient physical processes, governed by parabolic and hyperbolic PDEs, solution depends on both boundary data and initial conditions.