week2 - 1.7 The Lagrangian derivative (a.k.a. the...

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Unformatted text preview: 1.7 The Lagrangian derivative (a.k.a. the convective derivative, or the material derivative). We know how to measure the time derivative of a physical quantity associated with the fluid, for example that of the temperature T ( x ,t ), at a fixed point in space (the Eulerian derivative). Its just T t . This quantity will describe the change of temperature at a fixed location, for example air temperature in Bristol. However, this quantity will not be a measure of how a mass of air heats up or becomes colder. The reason is that air is swept away by the prevailing flow field u ( x ,t ). In other words, to describe the change of temperature of a piece of air, we need to consider the rate of change of T ( x ,t ), following a fluid particle . This is called the Lagrangian derivative. x+ x t+ t x=u t O x t Suppose a particle at time t is positioned at x , then at time t + t it has moved to x + u t . The change in T in the instant t is T ( x + u t,t + t ) T ( x ,t ) (6) Taylor expanding about x , t we get T ( x ,t ) + utT x ( x ,t ) + vtT y ( x ,t ) + wtT z ( x ,t ) + tT t ( x ,t ) + O (( t ) 2 ) T ( x ,t ) . The Lagrangian derivative is (6) divided by t as t 0, so that we get T t + ( u ) T DT Dt . (7) A particularly important example is the change in velocity (the acceleration) of a fluid particle, which we need to apply Newtons equations to fluid motion. Of course, the velocity is a vector quantity, which means we have to apply (7) to each component: D u Dt = parenleftbigg Du 1 Dt , Du 2 Dt , Du 3 Dt parenrightbigg . The the acceleration of a fluid particle becomes D u Dt = u t + ( u ) u . (8) Note: The Lagrangian derivative (i.e. following fluid particles) is given in terms of Eulerian (i.e. fixed point) measurements. It is vital to understand exactly how to com- pute expressions like ( u ) u for a given velocity. Example: Consider an accelerating fluid flow, such as the log flowing through a narrowing channel. Supposenarrowing channel....
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week2 - 1.7 The Lagrangian derivative (a.k.a. the...

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