Lab 1 - A Navier-Stokes Problem Statement of My Interest

Objectives

This first lab simply involves the statement of a CFD problem. A Navier-Stokes problem statement is given that is of my interst. This involves a summarization of the governing PDE system, a statement of its non-dimensionalization, and any appropriate boundary conditions.

Problem Statement

The Governing PDEs


The Navier-Stokes Problem of Interest

Starting with the beginning of this semester, I am currently transitioning into a doctorate program with Dr. Kenneth Kihm in the MABE department. My future research interests are very vague at this point but will generally concern micro-scale fluid flows. Therefore, for this problem of interest, I will give a general model for flow over a flat plate otherwise known as laminar boundary layer flow.

The simplifying assumptions of incompressible, two-dimensional, and steady-state flow allow for a tractable formulation of the Navier-Stokes equations. Furthermore, the assumption of constant fluid properties will decouple the momentum equation from the energy equation, further simplifying the problem.

The mass balance is of the form

which, with the above assumptions, simplifies to

The momentum balance in the most general form of the Navier Stokes equations is of the form

which, with the above assumptions, simplifies to two equations for the x and y dimensions.


The Non-Dimensionalization


Identifying Dimensionless Groups

The appropriate non-dimensionalization terms are listed below:



When applied to the mass equation,

the resulting form gives us an idea to the scale of the vertical velocity reference term:



Continuing with the non-dimensionalization of the Navier-Stokes equations yields the following for the x-dimension of the momentum equation

which reduces to

If we let recognize that the Reynolds number will be much larger than unity then

   since   

We thus observe that only one of the viscosity terms remains significant and we arrive at the same discovery made by Prandtl 101 year ago.



Following the same non-dimensionalization proceedure for the y-dimension of the momentum equation only tells us that the pressure gradient exists only in the x-direction.

Appropriate Boundary Conditions


Defining the Problem-space Boundaries

Since this is a second order partial differential equation in two dimensions, four boundary conditions are required to make the system tractable for solution: