where the last line is obtained using definitions of velocity components given

Where the last line is obtained using definitions of

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where the last line is obtained using definitions of velocity components given earlier in our discussion of streamlines (Chap. 2) and is identical to Eq. (3.1) except for notation on the left-hand side. We again emphasize that the substantial derivative of any property is simply an Eulerian- coordinate representation of the Lagrangian derivative of that property. Thus, in the case of velocity components the substantial derivative is the Lagrangian acceleration . This terminology is often used, but the term Eulerian acceleration is also sometimes employed with the same meaning. It is also important to observe that the D/Dt notation of Eq. (3.1), although the single most common one, is not found universally; in fact, the simple d/dt is also quite often employed, and sometimes termed “total acceleration.” If we take f = u , the x component of velocity, the substantial derivative given in Eq. (3.1) is Du Dt = ∂u ∂t bracehtipupleftbracehtipdownrightbracehtipdownleftbracehtipupright local accel. + u ∂u ∂x + v ∂u ∂y + w ∂u ∂z bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright convective acceleration , (3.2) and this represents the x -direction (Lagrangian, or total) acceleration, a x , of a fluid parcel expressed in an Eulerian reference frame. We see that this consists of two contributions. The first of these
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3.1. LAGRANGIAN & EULERIAN SYSTEMS; THE SUBSTANTIAL DERIVATIVE 51 is the local acceleration which would be present if we were to attempt to calculate acceleration only with respect to the Eulerian coordinates; it is simply the time rate-of-change of the velocity component u at any specified spatial location. The second is the set of terms, U · ∇ u = u ∂u ∂x + v ∂u ∂y + w ∂u ∂z , known as the convective acceleration . This quantity depends on both the local (in the same sense as above—point of evaluation) velocity and local velocity gradients, and it is the part of the total ac- celeration that arises specifically from the fact that the substantial derivative provides a Lagrangian description. In particular, it represents spatial changes in velocity (or any other fluid property) due to motion of a fluid parcel being carried (convected) by the flow field ( u, v, w ) T . It is important to observe that this contribution to the acceleration implies that fluid parcels may be accelerating even in a steady (time-independent) flow field, a result that might at first seem counterintuitive. Consider steady flow in a convergent-divergent nozzle shown in Fig. 3.3. As always, flow speed low-speed flow high-speed flow Figure 3.3: Steady accelerating flow in a nozzle. is indicated by the length of the velocity vectors, and from this we see that the flow is experiencing an increase in speed as it enters the converging section of the nozzle. Our intuition should suggest that this is likely to occur, and we will later be able to show, analytically, that this must be the behavior of incompressible fluids. The main point here, however, is the fact that the flow velocity is changing spatially even though it is everywhere independent of time. This in turn implies that the
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