Lagrangian Averaging

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Lagrangian averaging is directly related to the Lagrangian description of a system, which requires tracking the motion of each individual fluid particle. Therefore, Lagrangian averaging is a very useful tool when the dynamics of individual particles are of interest. To obtain Lagrangian time averaging, it is necessary to follow a specific particle and observe its behavior for a certain time interval. Then, the behavior of this particle is averaged over the time interval.

For a generalized function Φ = Φ(X,Y,Z,t),X,Y, and Z are material coordinates moving with the particle, and X,Y,Z are functions of the spatial coordinates x,y,z, and time t, i.e.,

\begin{array}{*{20}{c}}
   {X = X(x,y,z,t),} & {Y = Y(x,y,z,t),} & {Z = Z(x,y,z,t)}  \\
\end{array}

The most widely used Lagrangian averaging is time averaging, where the time average of the function Φ in time interval of Δt is

\bar \Phi  = \frac{1}{{\Delta t}}\int_{\Delta t} {\Phi (X,Y,Z,t)dt}     \qquad \qquad(1)

Lagrangian time averaging is performed for a distinct particle moving in the field; therefore, X,Y, and Z in the time interval Δt are not fixed in space. This focus on specific particles moving in space and time distinguishes Lagrangian averaging from Eulerian time averaging, which treats a fixed point in space relative to the reference frame. An example from daily experience will serve to illustrate this difference. In order to monitor traffic on the highway, the speed of all cars passing a point can be measured and averaged over a certain time interval – a case of Eulerian averaging. To catch an individual speeder, the police must follow the vehicle of interest to measure its speed as it moves in space over a certain time interval – a case of Lagrangian averaging.

References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA
Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Further Reading

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