f10_fall - Fluids Lecture 10 Notes 1 Substantial Derivative...

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Fluids – Lecture 10 Notes 1. Substantial Derivative 2. Recast Governing Equations Reading: Anderson 2.9, 2.10 Substantial Derivative Sensed rates of change The rate of change reported by a flow sensor clearly depends on the motion of the sensor. For example, the pressure reported by a static-pressure sensor mounted on an airplane in level flight shows zero rate of change. But a ground pressure sensor reports a nonzero rate as the airplane rapidly flies by a few meters overhead. The figure illustrates the situation. p (t) 1 p (t) 2 o t = t wing location at o t p (t) 1 p (t) 2 p t Note that although the two sensors measure the same instantaneous static pressure at the same point (at time t = t o ), the measured time rates are different. dp 1 dp 2 p 1 ( t o ) = p 2 ( t o ) but ( t o ) negationslash = ( t o ) dt dt Drifting sensor We will now imagine a sensor drifting with a fluid element . In effect, the sensor follows the element’s pathline coordinates x s ( t ), y s ( t ), z s ( t ), whose time rates of change are just the local flow velocity components dx s dy s dz s = u ( x s , y s , z s , t ) , = v ( x s , y s , z s , t ) , = w ( x s , y s , z s , t ) dt dt dt V p (t) s s p pathline pressure field sensor drifting with local velocity s Dp dp Dt dt t Consider a flow field quantity to be observed by the drifting sensor, such as the static pressure p (
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