AME 431/531

AME 431/531

Numerical Methods in Fluid Mechanics and Heat Transfer

This course prepares students for numerical solution of problems that involve fluid mechanics and/or heat
transfer. This topic is more widely known as Computational Fluid Dynamics (CFD). The course begins with
reviewing the fundamental concepts and continues with the derivation of the governing equations known as the
Navier-Stokes equations. Due to the nonlinearity and high dimensionality of these equations, numerical solutions
are needed, which will be covered in the remaining part of the course. To facilitate the grasp of the concepts, the
numerical methods are first discussed for simple linear equations and are then advanced toward the solution
methods for the full system of Navier-Stokes equations. Students will learn several widely-used numerical methods
and implement them in computer codes. The course includes two projects to give students experience with the
applications of the numerical methods to solve benchmark problems.

Prerequisite(s): AME 302 and AME 331

Project 1

Time Developing Flow in a Parallel Plate Channel

We consider the time evolution of a flow in a parallel plate channel. The fluid is considered initially be at rest, and then evolves to its steady state solution.

The governing Navier-Stokes equations for this flow may be simplified to:

Image

For this project, the student is expected to:

• State the boundary conditions for the problem
• Non-dimensionalize the boundary conditions and governing equation
• Mathematically determine the analytical solution
• Write a numerical finite-difference code in either Fortran or C/C++ using:
  • An explicit method
  • An implicit method
  • 4th-order Runge-Kutta method
• Investigate the following:
  • Accuracy considering the effects Δt, Δy, and r = Δt/(Δy)²
  • Stability considering the effects of Δt, Δy, and r

Sample Result

The video depicts the solution to this problem using an forward in time, central in space (FTCS), finite-difference scheme

Project Handout

The project handout may be downloaded here.

Project 2

Time Developing Flow in a Lid Driven Cavity

In this project, the student will develop a two-dimensional incompressible Navier-Stokes solver. The numerical method is based on the Marker and Cell approach by Harlow and Welch.

The code will then be used to solve a benchmark problem, the lid driven cavity, in order to verify its accuracy.

For this project, the student is expected to:

• State the boundary conditions for the problem
• Non-dimensionalize the boundary conditions and governing equations
• Write an explicit MAC method code in either Fortran or C/C++
• Compare with previously published results for the one-sided lid-driven cavity
• Investigate:
  • The effect of Reynolds number
  • Stability considering the effects of Δt and Δx = Δy

Sample Result

Here we see an example of the flow in a single-lid driven cavity at Reynolds number equal to 100. The solution is shown with velocity vectors and vorticity. Note that the solution is laminar and steady.

Benchmark Data

Validation data may be downloaded here:
Benchmark Validation Data [PDF]

Sample Result

Here we see an example of the flow in a single-lid driven cavity at Reynolds number equal to 10,000. Note that the velocity vectors are not steady, and a quasi-steady behavior emerges.

Project Handout

The project handout may be downloaded here