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  ==The Reynolds Averaged Equations and the Turbulence Closure Problem==
 
   
  '''1. The Equations Governing the Instantaneous Fluid Motions'''
 
 
 
  All fluid motions, whether turbulent or not, are governed by the dynamical equations for a fluid. These can be written using Cartesian tensor notation as:
 
 
 
  <table width="100%">
 
  <tr><td>
 
  :<math>
 
  \rho\left[\frac{\partial \tilde{u_i}}{\partial t}+\tilde{u_j}\frac{\partial \tilde{u_i}}{\partial x_j}\right] = \frac{\partial \tilde{p}}{\partial x_i}+\frac{\partial \tilde{T_{ij}}^{(v)}}{\partial x_j}</math>
 
  </td><td width="5%">(2.1)</td></tr></table>
 
  <table width="100%">
 
  <tr><td>
 
  :<math>
 
  \left[\frac{\partial \tilde{\rho}}{\partial t}+\tilde{u_j}\frac{\partial \tilde{\rho}}{\partial x_j}\right]+ \tilde{\rho}\frac{\partial \tilde{u_j}}{\partial x_j}= 0 </math>
 
  </td><td width="5%">(2.2)</td></tr></table>
 
 
 
 
 
  where <math>\tilde{u_i}(\vec{x},t)</math> represents the ithe component of the fluid velocity at a point in space,<math>[\vec{x}]_i=x_i</math>, and time,t. Also
 
  <math>\tilde{p}(\vec{x},t)</math> represents the static pressure, <math>\tilde{T_{ij}}^{(v)}(\vec{x},t)</math>, the viscous(or deviatoric) stresses, and <math>\tilde\rho</math> the fluid density. The tilde over the symbol indicates that an instantaneous quantity is being considered. Also the Einstein summation convention has been employed[1].
 
 
 
  In equation 2.1, the subscript i is a free index which can take on the values 1,2 and 3. Thus equation 2.1 is in reality three separate equations. These three equations are just Newton's second law written for a continuum in a spatial(or Eulerian) reference frame. Together they relate the rate of change of momentum per unit mass <math>(\rho{u_i})</math>,a vector quantity, to the contact and body forces.
 
 
 
  Equation 2.2 is the equation for mass conservation in the absence of sources(or sinks) of mass. Almost all flows considered in this material will be incompressible, which implies that derivative of the density following the fluid material[the term in brackets] is zero. Thus for incompressible flows, the mass conservation equation reduces to:
 
 
 
  <table width="100%">
 
  <tr><td>
 
  :<math>
 
  \frac{D \tilde{\rho}}{Dt}=\frac{\partial \tilde{\rho}}{\partial t}+\tilde{u_j}\frac{\partial \tilde{\rho}}{\partial x_j}= 0</math>
 
  </td><td width="5%">(2.3)</td></tr></table>
 
 
 
  From equation 2.2 it follows that for incompressible flows,
 
 
 
  <table width="100%">
 
  <tr><td>
 
  :<math>
 
  \frac{\partial \tilde{u_i}}{\partial x_j}= 0</math>
 
  </td><td width="5%">(2.4)</td></tr></table>
 
 
 
  The viscous stresses(the stress minus the mean normal stress) are represented by the tensor<math>\tilde{T_{ij}}^{(v)}</math>. From its definition,<math>\tilde{T_{kk}}^{(v)}</math>=0. In many flows of interest, the fluid behaves as a Newtonian fluid in which the viscous stress can be related to the fluid motion by a constitutive relation of the form.
 
 
 
  <table width="100%">
 
  <tr><td>
 
  <math>\tilde{T_{ij}}^{(v)}= 2\mu[\tilde{s_{ij}}\frac{1}{3}\tilde{s_{kk}}\delta_{ij}] </math>
 
  </td><td width="5%">(2.5)</td></tr></table>
 
 
 
  The viscosity, <math>\mu</math>, is a property of the fluid that can be measured in an independent experiment. <math>\tilde s_{ij}</math> is the instantaneous strain rate tensor defoned by
 
 
 
  <table width="100%">
 
  <tr><td>
 
  <math>\tilde{s_{ij}}= \frac{1}{2}\left[\frac{\partial \tilde u_i}{\partial x_j}+\frac{\partial \tilde u_j}{\partial x_i}\right] </math>
 
  </td><td width="5%">(2.6)</td></tr></table>
 
 
 
  From its definition, <math>\tilde s_{kk}=\frac{\partial \tilde u_k}{\partial x_k}</math>. If the flow is incompressible, <math>\tilde s_{kk}=0</math> and the Newtonian constitutive equation reduces to
 
  <table width="100%">
 
  <tr><td>
 
  <math>\tilde{T_{ij}}^{(v)}= 2\mu\tilde{s_{ij}}</math>
 
  </td><td width="5%">(2.7)</td></tr></table>
 
 
 
  Throughout this material, unless explicitly stated otherwise, the density <math>\tilde\rho=\rho</math> and the viscosity <math>\mu</math> will be assumed constant. With these assumptions, the instantaneous momentum equations for a Newtonian Fluid reduce to:
 
 
 
  <table width="100%">
 
  <tr><td>
 
  :<math>
 
  \left[\frac{\partial \tilde{u_i}}{\partial t}+\tilde{u_j}\frac{\partial \tilde{u_i}}{\partial x_j}\right] = \frac {1}{\tilde\rho}\frac{\partial \tilde{p}}{\partial x_i}+\nu\frac{\partial^2 {\tilde{u_i}}}{\partial x_j^2}</math>
 
  </td><td width="5%">(2.8)</td></tr></table>
 
 
 
  where the kinematic viscosity, <math>\nu</math>, has been defined as:
 
  <table width="100%">
 
  <tr><td>
 
  <math>\nu\equiv\frac{\mu}{\rho}</math>
 
  </td><td width="5%">(2.9)</td></tr></table>
 
 
 
  Note that since the density is assumed conastant, the tilde is no longer necessary.
 
 
 
  Sometimes it will be more instructive and convenient to not explicitly include incompressibilty in the stress term, but to refer to the incompressible momentum equation in the following form:
 
 
 
  <table width="100%">
 
  <tr><td>
 
  :<math>
 
  \rho\left[\frac{\partial \tilde{u_i}}{\partial t}+\tilde{u_j}\frac{\partial \tilde{u_i}}{\partial x_j}\right] = \frac{\partial \tilde{p}}{\partial x_i}+\frac{\partial \tilde{T_{ij}}^{(v)}}{\partial x_j}</math>
 
  </td><td width="5%">(2.10)</td></tr></table>
 
 
 
  This form has the advantage that it is easier to keep track of the exact role of the viscous stresses.
 
   
 ==2. Equations for the Average Velocity==   ==2. Equations for the Average Velocity== 