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 $(x, y) = (x<em>0, y</em>0)$.
• 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 $x<em>1$ in the solution region interior (0 < x1 < L) and time $t<em>1$ can only influence events at later times $(t > t</em>1)$.
• 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 $(x^<em>, t^</em>)$ depends on data at t < t^*.
• Elliptic PDE: Solution at point $(x^<em>, t^</em>)$ 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.