Descriptions of Fluid Flows

Descriptions of Fluid Flows

1. There are two ways to describe fluid flows:

   • The Lagrangian Description is one in which individual fluid particles are tracked
     • In the Lagrangian description of fluid flow, individual fluid particles are "marked," and their positions, velocities, etc. are described as a function of time.
       • If there were only a few particles to consider the Lagrangian description would be desirable.
   • The Eulerian Description is one in which a control volume is defined, within which fluid flow properties of interest are expressed as fields.
     • In the Eulerian description of fluid flow, individual fluid particles are not identified. Instead, a control volume is defined
       • In other words, each property is expressed as a function of space and time, as shown for the velocity field in the diagram.
         • Eulerian description of fluid flow, one is not concerned about the location or velocity of any particular particle, but rather about the velocity, acceleration, etc. of whatever particle happens to be at a particular location of interest at a particular time.
           • Example - Velocity field - An example of a fluid flow variable expressed in Eulerian terms is the velocity.
             • Note that since velocity is a vector, it can be split into three components (u,v, and w), all three of which are functions of space and time. In other words, at a given point in space (x,y, and z), and at some particular time (t), the velocity vector is defined. It is of no concern which fluid particle is at that location at that time. Whatever fluid particle happens to be at that location at time t has the velocity defined above.

2. The Material Derivative, also called the Total Derivative or Substantial Derivative is useful as a bridge between Lagrangian and Eulerian descriptions.

   • The material derivative of some quantity is simply defined as the rate of change of that quantity following a fluid particle. It is derived for some arbitrary fluid property Q as follows:
     • In this derivation, dt/dt = 1 by definition, and since a fluid particle is being followed, dx/dt = u, i.e. the x-component of the velocity of the fluid particle. Similarly, dy/dt = v, and dz/dt = w following a fluid particle.
       • Note that Q can be any fluid property, scalar or vector. For example, Q can be a scalar like the pressure, in which case one gets the material derivative or substantial derivative of the pressure. In other words, dp/dt is the rate of change of pressure following a fluid particle.
         • Thus, at some given spatial location (x,y,z) and at some given time (t), DQ/Dt = dQ/dt = the material derivative of Q, and is defined as the total rate of change of Q with respect to time as one follows whatever fluid particle happens to be at that location
           • The first term on the right hand side is called the local acceleration or the unsteady acceleration. It is only non-zero in an unsteady flow.
             • The last three terms make up the convective acceleration, which is defined as the acceleration due to convection or movement of the fluid particle to a different part of the flow field.
               • The convective acceleration can be non-zero even in a steady flow! In other words, even when the velocity field is not a function of time (i.e. a steady flow), a fluid particle is still accelerated from one location to another. Methods of Visualizing Fluid Flows

Images and files in the map