Fluid Mechanics - Revision history

VagifO: /* Numerical examples */

2024-03-02T13:23:09Z

‎Numerical examples

2023-12-27T12:32:07Z

2023-12-27T12:30:28Z

2022-10-22T19:39:18Z

‎Numerical examples

2019-08-27T09:23:13Z

‎Introduction

2019-05-18T15:05:04Z

‎Numerical examples

2018-09-22T10:31:31Z

2018-09-07T15:12:36Z

‎Artificial compressibility method

2018-08-30T08:33:21Z

2018-08-30T08:33:10Z

@@ Line 235: / Line 235: @@
 = Numerical examples =
 * [[Lid driven cavity]]
-* [[Burger's equation]]
+* [[Burgers' equation]]
 * [[de Vahl Davis natural convection test]]
 * [[Natural convection in 3D irregular domain]]

@@ Line 235: / Line 235: @@
 = Numerical examples =
 * [[Lid driven cavity]]
-* [[Burgers equation]]
+* [[Burger's equation]]
 * [[de Vahl Davis natural convection test]]
 * [[Natural convection in 3D irregular domain]]

@@ Line 235: / Line 235: @@
 = Numerical examples =
 * [[Lid driven cavity]]
 * [[de Vahl Davis natural convection test]]
 * [[Natural convection in 3D irregular domain]]

@@ Line 234: / Line 234: @@
 = Numerical examples =
 * [[Lid driven cavity]]
 * [[de Vahl Davis natural convection test]]
 * [[Natural convection in 3D irregular domain]]
 * [[Natural convection between concentric cylinders]]
+* [[Non-Newtonian fluid]]

@@ Line 7: / Line 7: @@
 \label{NavierStokes}
 \end{equation}
-also known as a Navier-Stokes equation. In many cases we are interested in the incompressible fluids (Ma<0.3), reducing the continuity equation to
+also known as a Navier-Stokes equation.
 Note that the $\b{v}\b{v}$ stands for the tensor or dyadic product \[ \b{v}\b{v} = \b{v}\otimes\b{v} = \b{v}\b{v}^\T = \left[ \begin{matrix}
@@ Line 21: / Line 14: @@
     {{v}_{n}}{{v}_{1}} & \cdots & {{v}_{n}}{{v}_{n}}  \\
 \end{matrix} \right]\]
-An example of incompressible variant of advection term in 2D would therefore be
 \[\left( \b{v}\cdot \nabla  \right)\b{v}=\left( \left( \begin{matrix}
     u  \\
@@ Line 36: / Line 39: @@
 \end{matrix} \right)\]
 The goal of CFD is to solve system \ref{NavierStokes} and \ref{contuinity}. It is obvious that a special treatment will be needed to couple both equations. In following discussion we cover some basic approaches, how this can be accomplished.
 = Solutions algorithms =

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 Otherwise an appropriate expression regarding the velocity can be written, i.e. write full  and taken in account velocity BCs. For example, Neumann velocity $\frac{\partial u}{\partial x}=0$ in 2D
 \[\frac{\partial p}{\partial x}=\left( \nu {{\nabla }^{2}}u + {{f}_{x}}-\frac{\partial u}{\partial t}+v\frac{\partial u}{\partial y} \right)\]
-Note that you allready know everything about the velocity and thus you can compute all the terms explicitely.
+Note that you allready know everything about the velocity and thus you can compute all the terms explicitly.

@@ Line 55: / Line 55: @@
 $C$ [m/s] - speed of sound
 \[\frac{1}{C^2}=\frac{\partial \rho }{\partial p}\]
-Or in another words
+Or in other words
 \[C^2=\left( \frac{\partial p}{\partial \rho}\right)_S\]
 where $\rho$ is the density of the material. It follows, by replacing partial derivatives, that the isentropic compressibility can be expressed as:

@@ Line 4: / Line 4: @@
 Long story short, we want to solve
 \begin{equation}
-\frac{\partial \b{v}}{\partial t}+\nabla \cdot ( \rho \b{vv}) = -\nabla p+\nu \nabla(\mu \nabla\b{v})+\b{f}
+\frac{\partial \b{v}}{\partial t}+\nabla \cdot ( \rho \b{vv}) = -\nabla p+ \nabla(\mu \nabla\b{v})+\b{f}
 \label{NavierStokes}
 \end{equation}