ENGR 314 - Session 03 Notes

Equilibrium and Marching Problems Recap

• Physical Processes: Can be classified as either equilibrium or marching problems.

• Equilibrium Problems:

  • Steady-state problems like temperature distribution in a solid rod, stress distribution in a loaded structure, and steady fluid flows.

  • Governed by elliptic PDEs.

• Marching Problems:

  • Include transient heat transfer, unsteady flows, and wave phenomena.

  • Governed by parabolic or hyperbolic PDEs.

Hyperbolic PDEs

• Occurrence: Appear in time-dependent processes with minimal energy dissipation, such as small amplitude vibrations and sound propagation.

• Model Equation: Wave equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ (Equation 3 in the slides), where $c$ is the wave speed.

• Influence: Disturbances at a point affect only a limited region in space.

• Problem Type: Yield initial-boundary-value problems.

• Applications: Crucial for solving and studying shock discontinuities in transonic and supersonic flows (e.g., aircraft aerodynamics, gas turbines).

Solving the Wave Equation

• Wave Equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ which is equation (3)

• Variable Transformations:

  • Using transformations ${\xi = x + ct, \eta = x - ct}$

  • Applying the chain rule repeatedly, the PDE becomes: $\frac{\partial^2 u}{\partial \xi \partial \eta} = 0$ which is equation (4)

• Solution: The general solution of PDE (4) is $u(x,t) = F<em>1(x + ct) + F</em>2(x - ct)$ which is equation (5), where $F<em>1$ and $F</em>2$ are arbitrary functions/ simple wave solutions

Characteristics of the Wave Equation

• Traveling Waves: $F<em>1$ and $F</em>2$ represent traveling waves with speed $c$ that propagate without changing shape or amplitude.

• Equation (5) Implications:

  • In the x-t plane:

    • $F_1$ is constant along lines of slope $dt/dx = 1/c$.

    • $F_2$ is constant along lines of slope $dt/dx = -1/c$.

  • Characteristics: These lines are called characteristics.

D’Alambert’s Solution to the Wave Equation

• Dependence on Initial Conditions: Functions $F<em>1$ and $F</em>2$ depend on the initial conditions of the function $\\phi$.

• Enforcing Initial Conditions: The complete solution is:
  $\phi(x, t) = \frac{1}{2} [\phi(x + ct) + \phi(x - ct)] + \frac{1}{2c} \int_{x-ct}^{x+ct} \psi(s) ds$ , which is equation (8)

• D’Alambert’s Finding: The value of $\\phi$ at point $(x, t)$ depends only on the data within the interval $(x - ct, x + ct)$, including the interval extremes.

Domain of Dependence

• Characteristics Intersection: Characteristics through point $(x′, t′)$ intersect the x-axis at points $(x′ - ct′, 0)$ and $(x′ + ct′, 0)$.

• Zone of Influence: Changes at point $(x′, t′)$ influence events at later times within the zone of influence, bounded by characteristics.

• Domain of Dependence Defined: The region in the x-t plane enclosed by the x-axis and the two characteristics for t < t′.

• Solution Dependence: The solution at $(x′, t′)$ is influenced only by events inside the domain of dependence.

Domain of Dependence of Different PDE Types

• Hyperbolic: Domain of dependence is bound by characteristics.

• Elliptic: Domain of dependence encompasses the entire domain.

• Parabolic: Domain of dependence extends infinitely in one direction.

Example: Localized Disturbance

• Initial Condition:
  $u(x, 0) = f(x) = \begin{cases} 1, &amp; \text{if } -3 \leq x \leq 3 \ 0, &amp; \text{otherwise} \end{cases}$
  $\frac{\partial u}{\partial t}(x, 0) = g(x) = 0$

• Interpretation: initial disturbance can be seen as pressure spike, wave crest, temperature surge, etc.

• Solution: $u(x, t) = \frac{1}{2} [f(x + ct) + f(x - ct)]$

Example: Left and Right Propagation

• Disturbance Splitting: For t > 0, the initial disturbance splits into two parts, one propagating to the left and the other to the right.

Another Hyperbolic Example: Sound Propagation

• Speed of Sound: $\Delta x = c \Delta t$, where $\Delta x$ is the distance, $c$ is speed of sound and $\Delta t$ is delta time.

Key Links Between Model and CFD PDEs

• Time-Dependent Navier-Stokes (NS) Equations:

  • Form a system of parabolic PDEs.

  • Solved as Initial Boundary Value Problems (IBVP).

  • Require initial and boundary conditions.

• Compressible Time-Dependent Euler Equations:

  • Obtained by removing viscous terms from NS equations.

  • Form a system of hyperbolic PDEs.

• Steady NS Solutions:

  • Form a system of elliptic PDEs.

  • Solved as the asymptotic state of an unsteady problem, requiring initial data.

• CFL (Courant-Friedrichs-Lewy) Constraints:

  • Time-step constraints of both NS and Euler CFD analyses are understood through the characteristics of wave equations.

Summary

• Characteristics of Wave Equation: Discussed characteristics as a model PDE for the hyperbolic marching problem.

• Information Travel: Information travels unaltered along characteristics of hyperbolic PDEs.

• D’Alambert Solution: Discussed D’Alambert’s solution to the wave equation.

• Physical Interpretation: Introduced characteristics of wave equation and their physical interpretation.

• CFL Time-Step Constraint: Characteristics of the wave equation will be necessary to explain the physical meaning of the Courant-Friedrichs-Lewy (CFL) time-step constraint of Navier-Stokes CFD codes.

Appendix 1: Derivation of D’Alambert’s Solution

• Introducing new variables: ${\nu = x + ct, \zeta = x - ct}$

Let $\mu(x, t) = u(\nu, \zeta)$

Using these new variables, the derivative with respect to x & t can be rewritten as:

$\frac{\partial u}{\partial x} = \frac{\partial u}{\partial \nu} \frac{\partial \nu}{\partial x} + \frac{\partial u}{\partial \zeta} \frac{\partial \zeta}{\partial x} = \frac{\partial u}{\partial \nu} + \frac{\partial u}{\partial \zeta}$

$\frac{\partial u}{\partial t} = \frac{\partial u}{\partial \nu} \frac{\partial \nu}{\partial t} + \frac{\partial u}{\partial \zeta} \frac{\partial \zeta}{\partial t} = c \frac{\partial u}{\partial \nu} - c \frac{\partial u}{\partial \zeta}$

• Similarly,

$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial \nu^2} - 2c^2 \frac{\partial^2 u}{\partial \zeta \partial \nu} + c^2 \frac{\partial^2 u}{\partial \zeta^2}$

By converting all derivatives in x & t into derivatives in $\nu$ & $\zeta$, the wave equation becomes:

$\frac{\partial^2 u}{\partial \zeta \partial \nu} = 0$

This equation can be integrated twice:

$u(\nu, \zeta) = \phi(\nu) + \psi(\zeta)$

$\therefore u(x, t) = \phi(x + ct) + \psi(x - ct)$

Appendix 1 - cont'd

• If given the initial conditions
  $u(x,t=0) = f(x)$
  $\frac{\partial u}{\partial t}(x,t=0) = g(x)$

• Determine the D'Alambert's solution:
  $u(x, 0) = \phi(x+ct) + \psi(x-ct) = \phi(x) + \psi(x) = f(x)$
  $\frac{\partial}{\partial t} u(x,t) = c\phi'(x+ct) - c\psi'(x-ct)$
  $\frac{\partial}{\partial t} u(x,t=0) = c[\phi'(x) - \psi'(x)] = g(x)$
  $\therefore u(x,t) = \frac{1}{2}[f(x+ct)+f(x-ct)] + \frac{1}{2c} \int_{x-ct}^{x+ct} g(s) ds$

Appendix 2: Characteristics

If we assume $\frac{\partial u}{\partial t}(x)=0$ for simplicity
$u(x,t) = \frac{1}{2}[f(x+ct)+f(x-ct)]$
Specify x + ct = constant, therefore f(x+ct) remains the same as long as x + ct remains a constant.

Appendix 3

Not all marching problems are unsteady. Certain steady flows can also be treated as marching problems.

In these cases, flow direction acts as a time-like coordinate along which marching is possible.

In these cases, problem dimensionality is reduced by factor 1, with significant computational saving.

Examples: inviscid supersonic flows (hyperbolic) and viscous supersonic flows (parabolic).