Chapter 15, Problem 1CP

A CFD code is used to solve a two-dimensional (x and y), incompressible, laminar flow without free surfaces. The fluid is Newtonian. Appropriate boundary conditions e used. List the variables (unknowns) in the problem, and list the corresponding equations to be solved by the computer.

To determine

The list of variables and corresponding equation used to solve on the computer.

Explanation of Solution

According to the data given in question for an incompressible Newtonian laminar flow fluid with no free surface. The unknown quantity present in the Stokes flow equation are u, v and P.

The corresponding equation for the given flow condition is continuity of Navier-Stokes equation. The other equation is: -

1. X- component of Navier-Stokes equation
2. Y-component of Navier-Stokes equation

Navier-Stokes equation in Cartesian co-ordinate system is given as:

x-component:

  $\rho \left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)=-\frac{\partial P^{\prime }}{\partial x}+\mu {\nabla }^{2}u$

y-component of Navier-Stokes equation:

  $\rho \left(\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\right)=-\frac{\partial P^{\prime }}{\partial y}+\mu {\nabla }^{2}v$

z-component:

  $\rho \left(\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\right)=-\frac{\partial P^{\prime }}{\partial z}+\mu {\nabla }^{2}w$

Now,

During, non-dimensionalization a modified pressure is introduced. This helps in absorbing the hydrostatic pressure. It is defined as:

  $\text{<msup><mi>P</mi><mo>′</mo></msup>}\text{=}\text{P}\text{+}\text{ρgz}$

Here,

P is pressure,

  P′ is modified pressure

Z is the vertical direction vector

  $\text{ρ}$ is the density of fluid

When these equations are combined with appropriate boundary condition the problem can be solved.