Wake drag of moving/stationary flat plates: Not identical?

[A.T.]

I feel confused but from what I understand so far from you is that in both of stationary plate or moving plate scenarios, there is only ONE single reason for wake drag or back-of-plate-drag which is: the air rushing from the front over and around the edges?

Reasoning based on reasons is often not very reasonable, when it comes to fluid dynamics. See the endless discussions about the reason of lift. Sometimes the reason you can reasonably give for something might even depend on the frame of reference.

 

[Chestermiller]

The figure below presents a sketch of the streamline pattern for the case in which the plate spans the channel between the walls, and is shown as reckoned from the rest frame of the plate (i.e.,an observer moving with the plate).

In this frame of reference, the plate is stationary, and the walls are sliding to the right with a velocity V in the positive x direction. To the left of the plate, fluid is dragged (by the walls) to the right (toward the plate) near the top and bottom of the channel, and returns to the left along the middle of the channel. The net flow is zero. To the right of the plate, fluid is dragged (by the walls) to the right (away from the plate) near the top and bottom of the channel, and flows to the left along the middle of the channel. Again, the net flow is zero.

This kind of flow is well known in polymer processing operations involving screw pumps and screw extruders. The plate in our system assumes the role of the screw flight in an extruder. We can employ the same type of approach as we use for extruders to solve for the flow and pressure distribution in our system. The flow is a combination of "pressure flow" and "drag flow," with the net flow being zero. For the case of laminar flow of a viscous fluid, the fluid velocity (at distances greater than about 1 plate height on either side of the plate) is essentially horizontal, and given by:
$$v_x=V\left[\frac{3(\frac{y}{h})^2-1}{2}\right]$$
where y is the distance measured upward from the centerline of the channel and h is half the height of the plate. The pressure gradient along the channel is given by
$$\frac{dp}{dx}=\frac{3V\eta}{h^2}$$
where $\eta$ is the viscosity. This equation applies on both sides of the plate. Across the plate itself, there is a discontinuous drop in pressure from the left side of the plate to the right side $\Delta p$. To the left of the plate, the pressure is higher than atmospheric, and to the right, the pressure is less than atmospheric. The pressure drop across the plate is given by:
$$\Delta p=\frac{3V\eta}{h^2}L$$
where L is the length of the tunnel.

 

[boneh3ad]

The figure below presents a sketch of the streamline pattern for the case in which the plate spans the channel between the walls, and is shown as reckoned from the rest frame of the plate (i.e.,an observer moving with the plate).
View attachment 94096
In this frame of reference, the plate is stationary, and the walls are sliding to the right with a velocity V in the positive x direction. To the left of the plate, fluid is dragged (by the walls) to the right (toward the plate) near the top and bottom of the channel, and returns to the left along the middle of the channel. The net flow is zero. To the right of the plate, fluid is dragged (by the walls) to the right (away from the plate) near the top and bottom of the channel, and flows to the left along the middle of the channel. Again, the net flow is zero.

I've given this some thought and I actually don't think this is a good analog to his example with the plate filling the whole tunnel. The idea with the plate filling the tunnel woudl be that some incoming uniform flow reaches the plate. At that point, whose to say whether the flow "turns backward" near the walls or near the centerline (i.e. which direction does the moving pipe actually move)? Further, if you have the plate fully blocking the cross-section, then why would there be any air movement on both sides as predicted by this model problem?

I'd instead argue that the correct model problem is simply a static problem where one side has a different static pressure than the other and the air is essentially not moving. If the fan is upstream (uncommon in a wind tunnel) then the upstream side of the plate will have a higher pressure from the fan compressing it slightly, but no air movement because there is simply nowhere for the air to go. Downstream of the plate would just be atmospheric. If the fan is downstream (almost always the configuration in a wind tunnel) then the upstream region would be atmospheric and the downstream region would be at a slight vacuum as the fan tries to pull air out but can only do so much with the tunnel closed off.

 

[Chestermiller]

I've given this some thought and I actually don't think this is a good analog to his example with the plate filling the whole tunnel. The idea with the plate filling the tunnel woudl be that some incoming uniform flow reaches the plate. At that point, whose to say whether the flow "turns backward" near the walls or near the centerline (i.e. which direction does the moving pipe actually move)? Further, if you have the plate fully blocking the cross-section, then why would there be any air movement on both sides as predicted by this model problem?

I'd instead argue that the correct model problem is simply a static problem where one side has a different static pressure than the other and the air is essentially not moving. If the fan is upstream (uncommon in a wind tunnel) then the upstream side of the plate will have a higher pressure from the fan compressing it slightly, but no air movement because there is simply nowhere for the air to go. Downstream of the plate would just be atmospheric. If the fan is downstream (almost always the configuration in a wind tunnel) then the upstream region would be atmospheric and the downstream region would be at a slight vacuum as the fan tries to pull air out but can only do so much with the tunnel closed off.

Maybe I didn't make myself clear. In the problem the OP posed, the plate is moving down the tunnel, and fills the channel, while the walls are stationary. From the frame of reference of the plate, this is the same as the plate being stationary, and the walls of the channel moving. There is no need for a fan to produce any flow in this situation. The movement of the channel walls drags air through. In this frame of reference, there is no flow past the stationary plate. The velocities of both the walls and fluid in this situation can be used to determine the velocities in the OP's problem statement simply by adding a constant velocity of V in the negative x direction to the fluid, the plate, and the walls. In that frame of reference, the walls are fixed and the plate is moving in the negative x direction with velocity V, while the fluid at the plate is moving with the plate velocity. So, what I'm saying is that in the stationary plate frame of reference, there is no need for a fan.

 

[boneh3ad]

Maybe I didn't make myself clear.

Nope, you were clear. I just didn't adequately read the two situations proposed by the OP in his more recent post apparently.

 

[Chestermiller]

Nope, you were clear. I just didn't adequately read the two situations proposed by the OP in his more recent post apparently.

Well, his Case B can't be used to represent Case A in a different reference frame.

Chet