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

[leviterande]

Hi All!

See the picture.
A-Consider a flat plate physically moving perpendicularly through the air.

B-Consider this same flat plate sitting perpendicularly and stationary now in a wind tunnel where air is made to flow.
Accordingly if airspeeds, areas etc are similar, the 2 situations should produce the exact same results.

However, I am interested in comparing only the rear wake drag in the two situations.

In situation A, if I understand correctly, main cause of the rear wake drag comes mostly from the the air being displaced ahead by the moving plate thus creating and leaving a rarified partial vacuum area behind the plate which wants to suck air from top/bottom side of the plate which results in the negative drag pressure i.e. wake drag on the plate.

Now, in situation B, the air coming from the front is similarly hitting the front of the plate and similarly basically not easily allowed to pass behind the plate. BUT!, in situation B there is still ambient air behind the plate, i.e. there is no rarified air or partial vacuum behind the plate because the plate in sitiation B hasnt physically displaced any air mass, right?

I thought perhaps too much about this issue which may have led to this very unintelligent question, still I would love to have clarification.

So, are the wake drags of A and B yet exactly equal? If so, may you explain how the -identical-to-A- wakedrag of B is caused?

Thanks

 

[Chestermiller]

Hi All!

See the picture.
A-Consider a flat plate physically moving perpendicularly through the air.

B-Consider this same flat plate sitting perpendicularly and stationary now in a wind tunnel where air is made to flow.
Accordingly if airspeeds, areas etc are similar, the 2 situations should produce the exact same results.

However, I am interested in comparing only the rear wake drag in the two situations.

In situation A, if I understand correctly, main cause of the rear wake drag comes mostly from the the air being displaced ahead by the moving plate thus creating and leaving a rarified partial vacuum area behind the plate which wants to suck air from top/bottom side of the plate which results in the negative drag pressure i.e. wake drag on the plate.

Now, in situation B, the air coming from the front is similarly hitting the front of the plate and similarly basically not easily allowed to pass behind the plate. BUT!, in situation B there is still ambient air behind the plate, i.e. there is no rarified air or partial vacuum behind the plate because the plate in sitiation B hasnt physically displaced any air mass, right?

I thought perhaps too much about this issue which may have led to this very unintelligent question, still I would love to have clarification.

So, are the wake drags of A and B yet exactly equal? If so, may you explain how the -identical-to-A- wakedrag of B is caused?

Thanks

Just imagine what you would see if you were moving along with the same velocity as the plate. It is exactly the same thing you would see if both you and the plate were stationary in a wind tunnel.

 

[boneh3ad]

Also, be careful with the term "rarefied". It has a very specific meaning in the context of fluid mechanics and typically denotes a region of fluid where the mean free path is large enough that the continuum assumption breaks down.

 

[leviterande]

Just imagine what you would see if you were moving along with the same velocity as the plate. It is exactly the same thing you would see if both you and the plate were stationary in a wind tunnel.

Sorry, perhaps the word rarefied was not adequate, what I mainly meant was that the air flow directly behind the moving plate seemed different than behind the stationary plate.

Ok, let's see, I am moving with the moving plate, let's say I am at the center of the plate area and looking now downstream towards the center of the wake. What I see is that the air particles are moving away from me further downstream, where--directly behind the plate- there is a zone of lower pressure created by the plate as it has scooped out the air. In other words, immediately behind the plate and at the center of the wake there are less air particles than far far downstream.I am now sitting at the center of the stationary plate in the windtunnel, The windtunnel is still Off, I look at the space where the wake will be created and I see stationary airparticles at ambient pressure. The fan of the windtunnel starts, those air particles immediately behind the plate at the center of the wake are still
stationary(?), as the plate itself hasn't scooped out any air. And, well, you see, that is where my problem lies.

Attempt at understanding
The only way I can imagine how the air particles in the center of the wake directly behind the stationary plate are affected is by the interaction of escaping airstreams coming from the front and out from the plate side edges. We could see that the viscosity of these edge air streams are affecting the air directly behind the plate. Sure, but this viscosity effect also exist in the moving plate scenario A in addition to the displaced air effect and thus is not an adequate explanation In my humble opinion? Thanks

 

[boneh3ad]

The two situations are identical. What @Chestermiller was attempting to have you do is imagine the plate moving me through the air. You are looking at it from the side like your picture shows. Now imagine while it moves, you are moving the same rate still looking at it.

 

[leviterande]

(Of course I realize that the two wakes of the two situations ought to be identical somehow, but how? )

I have considered and imagined several frames of reference. But still I don't understand how the air in the wakes would be similar. I outlined my thinking in the earlier posts.

 

[boneh3ad]

Well both instances must start from rest, correct? In both cases then there will initially be no pressure difference between the front and the back. Now, whether it is the air that starts moving or the plate, that fact doesn't change, yet in at least one instance according to your line of reasoning, a low-pressure region develops behind the plate. It must be that this low pressure region develops due to the motion, and the motion in the two situations are identical except for a simple Galilean transformation. So they start from the same condition and end with the same free stream condition so the fields should be the same.

(Apologies if this isn't clear. I'm sneaking typing while my wife shops.)

 

[leviterande]

Well both instances must start from rest, correct? In both cases then there will initially be no pressure difference between the front and the back. Now, whether it is the air that starts moving or the plate, that fact doesn't change, yet in at least one instance according to your line of reasoning, a low-pressure region develops behind the plate. It must be that this low pressure region develops due to the motion, and the motion in the two situations are identical except for a simple Galilean transformation. So they start from the same condition and end with the same free stream condition so the fields should be the same.

(Apologies if this isn't clear. I'm sneaking typing while my wife shops.)

Your words are clear. Obviously though, the subject is more clear to you than it is to me, a fact which may make you bypass some things that may be regarded as straightforward to you but not to me:). Happy shopping btw, (hope its not too heavy on the wallet hehe).

In both cases then there will initially be no pressure difference between the front and the back. Good.

Now, whether it is the air that starts moving or the plate, that fact doesn't change, What exact fact were you referring to here? what fact doesn't change do you mean? the fact of "no pressure difference"?

yet in at least one instance according to your line of reasoning, a low-pressure region develops behind the plate Do you agree that there is a low pressure behind?

It must be that this low pressure region develops due to the motion, and the motion in the two situations are identical except for a simple Galilean transformation. Yes and perhaps here I can come closer to the heart of the understanding issue. If I put it this way:

My question concerns specifically the flow immediately behind the stationary plate.
My reasoning is that as the wind tunnel fan starts, this same airflow will for the most part not be able to get behind the plate, thus my "erroneous" reasoning implies that the space with its air mass behind the plate can thus not be affected in anyway i.e. no motion of the air behind the plate and no low pressure behind the plate. If my point is not clear I will make a drawing of another better example.

 

[Vedward]

I disagree with the description of the moving & stationary plate cases. In both cases, air is being forced from the leading face of the plate off the sides. There is no vacuum formed on the back side, air doesn't jump from the edges to move behind the plate, but swirls form based properties of the air flow at the edges, these are either carried away by the flowing air or stay in place as the plate moves away. The effect looks the same for both moving plate & stationary plate in wind tunnel.

 

[Chestermiller]

Hi leviterande,

Here's a thought experiment for you.

You're in a tunnel. You're sitting on a cart, and the flat plate is mounted facing forward on a vertical pole affixed to the cart. The cart can move forward with a velocity V, and the ride is so smooth that you can't even feel the cart moving. The walls and floor of the tunnel are so smooth that you can't tell visually whether the cart is moving forward or not. There is a fan ahead of you in the tunnel that is capable of blowing air at a uniform velocity V toward you. The fan is either far enough away or is transparent, so that you can't see it.

Situation 1: The cart is moving forward with velocity V, but the fan is not blowing
Situation 2: The cart is stationary, and the fan is blowing air toward you with velocity V.

In both situations, the velocity of the air relative to the cart is V, and the velocity of the cart relative to the air is V (in the other direction).

I submit that there is no experiment you can do to prove that you are experiencing Situation 1 (and not Situation 2), or vice versa. If you can think of one, please describe it. The air velocities that you measure from your frame of reference on the cart will be exactly the same, the streamlines will be exactly the same, and the pressures at various locations (including behind the plate) will be exactly the same.

Chet

 

[leviterande]

I appreciate your replies. I understand that the 2 situations should be the same, -a fact you have made clear- I just don't grasp yet the how; how the two wakes are identical. it seems my exact question wasnt clear.

My current understanding of negative pressure drag

Besides the eddies behind the 2 plates creating drag I imagined there is another source of drag existing only in the moving plate:
We take the wind tunnel wall as the frame of reference. As the plate physically moves forward, the air in the space behind it must accelerate trying to catch up/fill up the momentarily hole/vacuum that the plate created behind. Since obviously air cannot accelerate instantly, and since the plate is constantly moving, there will be a low pressure zone behind the moving plate where the air behind tries to fill up the void. How wrong Am I in this thinking ?
Regards
Levi.

 

[Chestermiller]

This is exactly the same thing that happens with the wind blowing past the stationary plate.

 

[leviterande]

I feel that what I wanted to know, the issue in question, seems -despite what I wrote- to not have came out clear.
I understand that the situations should be the same but I want to understand how, how the air directly behind is the same for the 2 situations.
To illustrate my point even further look at the picture below, replace the flat plate with a C- shaped plate, i.e. the flat plate now has sides on top and bottom. For simplicity's sake let's assume from now that all flows are 2D.
The moving C plate has a low pressure air behind it as described earlier.
Now, how will this low pressure form behind the stationary plate? The air from the fan of the wind tunnel can barely get around and behind the C plate?

 

[Chestermiller]

I feel that what I wanted to know, the issue in question, seems -despite what I wrote- to not have came out clear.
I understand that the situations should be the same but I want to understand how, how the air directly behind is the same for the 2 situations.
To illustrate my point even further look at the picture below, replace the flat plate with a C- shaped plate, i.e. the flat plate now has sides on top and bottom. For simplicity's sake let's assume from now that all flows are 2D.
The moving C plate has a low pressure air behind it as described earlier.
Now, how will this low pressure form behind the stationary plate? The air from the fan of the wind tunnel can barely get around and behind the C plate?

I don't see how you can say that. The air in the stationary C plate picture is also blowing around the outside of the plate, and circles around in back and down.

I would like to see what you think the streamlines look like for the stationary C plate case. Can you please prepare a sketch of what you think the streamlines look like. (I know what they look like, but I'd like to give you a chance to work it out).

Chet

 

[Vedward]

My final comment on this: Unless the moving plate is moving at the speed of sound, there will not be a low pressure region behind the plate. If you put Pitou tubes on the front, back and side edges of the plate, the static pressure will be the same for all three locations. This also is the case for the wind tunnel plate.

However, at supersonic speeds, funny things start to happen. That is why rocket nozzles get larger in the direction of flow, this forces the exhaust to increase speed.

 

[Chestermiller]

My final comment on this: Unless the moving plate is moving at the speed of sound, there will not be a low pressure region behind the plate. If you put Pitou tubes on the front, back and side edges of the plate, the static pressure will be the same for all three locations. This also is the case for the wind tunnel plate.

This is definitely not correct. If this were the case, then there would be uniform pressure at the cylinder surface in both viscous and inviscid fluid flow past a cylinder. We know that that is not the case.

 

[boneh3ad]

My final comment on this: Unless the moving plate is moving at the speed of sound, there will not be a low pressure region behind the plate. If you put Pitou tubes on the front, back and side edges of the plate, the static pressure will be the same for all three locations. This also is the case for the wind tunnel plate.

However, at supersonic speeds, funny things start to happen. That is why rocket nozzles get larger in the direction of flow, this forces the exhaust to increase speed.

If viscosity was nonexistent then you'd be onto something. Since fluids do have viscosity, however, this is not correct. There will be a massive low pressure region behind the plate.

 

[A.T.]

My reasoning is that as the wind tunnel fan starts, ...

The Galilean invariance argument applies for constant velocities (two inertial frames): A plate that has moved at constant speed for a while has the same flow as a plate in a wind tunnel which has run at constant speed for a while.

However, in the transition phase, when either the plate or the air is accelerating, there is no equivalence and the flow might look different. For example, in the frame of the accelerating plate you have an inertial force field, which isn't there in the frame of the plate in the wind tunnel.

 

[leviterande]

Indeed, in accelerating frames things get different. However I am luckily only talking about constant speeds:)(at least I think I am thus far hehe)

I feel that my exact point was probably not clear so I am going to use an extreme example (possibly inappropriate but should show the point ) that I originally wanted to use at first but due to some reasons I didn't. Ok, let's assume that the flat plate and it´s two cases still occur in the same single wind tunnel. This flat plate however is so big that it fits exactly the cross section of the tunnel. The flat plate in other words is free to glide back and forth on air tight bearings through the tubular wind tunnel. Air from front/behind of the plate can't get past the plate. The wind tunnel is very long but is opened at both ends. In front of the plate we have the same fan.

Case A
Fan is off, plate is physically moving in a direction towards the fan.
There will be positive pressure ahead acting on the front of the plate
Furthermore, there will ALSO be negative pressure behind acting on the rear of the plate

Case B
Fan is on, Plate is stationary.
There will be positive pressure ahead acting on the front of the plate
Here however, the way I see it, we don´t have negative pressure behind acting on the rear of the plate. The air behind is not affected by the stream coming from the fan.

I appreciate your patience

 

[Chestermiller]

There is a stationary plate version of Case A that gives the exact same pressure distribution, but does not involve a fan. Can you figure out what that is?

 

[A.T.]

This flat plate however is so big that it fits exactly the cross section of the tunnel.

Here again the Galilean equivalence used in wind tunnels stops being applicable. The idea of a wind tunnel is that you have only two objects relevant for the flow:
- The test body (here the plate)
- The airmass
Galilean invariance tells you that it doesn't matter which of the two is moving.

In your scenario the boundaries of the tunnel become relevant for the flow, as a third object. There is no Galilean Transformation that makes the two cases equivalent, with that third object involved.

 

[Vedward]

Case A is a bicycle tire pump. The air in front of the plate gets hot because it is being compressed, the air behind the plate is atmospheric, no change in temperature and it is not in a vacuum. The air just follows along with the back of the plate.

 

[Chestermiller]

Here again the Galilean equivalence used in wind tunnels stops being applicable. The idea of a wind tunnel is that you have only two objects relevant for the flow:
- The test body (here the plate)
- The airmass
Galilean invariance tells you that it doesn't matter which of the two is moving.

In your scenario the boundaries of the tunnel become relevant for the flow, as a third object. There is no Galilean Transformation that makes the two cases equivalent, with that third object involved.

Hi A.T.,

Did you see my post #20? The actually is a Galilean Transformation that works for Case A, but it doesn't involve a fan.

Chet

 

[Vedward]

One more item, for a flat plate, the air will not curl around the edges. In both cases the air is forced to move parallel to the plate, once it reaches the edge, it continues in a straight line, due to the change in momentum caused by the flat plate changing the direction of flow. Other shapes (cylinder, square, sphere, etc) will have different flows.

Flat plate moving or stationary with air moving against it will have the same flow patterns, no negative pressure behind it.

 

[Chestermiller]

One more item, for a flat plate, the air will not curl around the edges. In both cases the air is forced to move parallel to the plate, once it reaches the edge, it continues in a straight line, due to the change in momentum caused by the flat plate changing the direction of flow. Other shapes (cylinder, square, sphere, etc) will have different flows.

Flat plate moving or stationary with air moving against it will have the same flow patterns, no negative pressure behind it.

Please provide a diagram of what you think the streamline pattern looks like for air blowing past a stationary flat plate (illustrating what you said above).

 

[A.T.]

for a flat plate, the air will not curl around the edges. In both cases the air is forced to move parallel to the plate, once it reaches the edge, it continues in a straight line

 

[Chestermiller]

Case A is a bicycle tire pump. The air in front of the plate gets hot because it is being compressed, the air behind the plate is atmospheric, no change in temperature and it is not in a vacuum. The air just follows along with the back of the plate.

This is not correct. The tunnel is very long and open at both ends, which is very different from compression of a gas in a cylinder. The flows and deformations are entirely different. Any air that is heated near the plate is swept away (forward) relative to the plate.

Please provide a diagram of what you think the streamline pattern for this situation looks like in the stationary-plate Galilean reference frame.

Chet

 

[boneh3ad]

One more item, for a flat plate, the air will not curl around the edges. In both cases the air is forced to move parallel to the plate, once it reaches the edge, it continues in a straight line, due to the change in momentum caused by the flat plate changing the direction of flow. Other shapes (cylinder, square, sphere, etc) will have different flows.

Flat plate moving or stationary with air moving against it will have the same flow patterns, no negative pressure behind it.

You can keep repeating this as many times as you want, but that doesn't make it true. You can find an incredibly large quantity of evidence refuting this claim if you, for example, do a Google search.

If Google isn't your style, how about a search in the Journal of Fluid Mechanics, which is arguably the most respected journal in the field. (EDIT: It turns out that the link here just goes to the article listing and doesn't actually link to the search. You can do the search yourself by just typing "normal flat plate" into the search field).

Not a fan of JFM? How about a search in Physics of Fluids, arguably the second most respected journal in the field.

 

[A.T.]

You can keep repeating this as many times as you want, but that doesn't make it true. You can find an incredibly large quantity of evidence refuting this claim if you, for example,

@Vedward cold also do some trivial experiments:

 

[boneh3ad]

Case B
Fan is on, Plate is stationary.
There will be positive pressure ahead acting on the front of the plate
Here however, the way I see it, we don´t have negative pressure behind acting on the rear of the plate. The air behind is not affected by the stream coming from the fan.

Why do you think the air behind is not affected by the stream coming from the fan? Consider the case of a wind tunnel where you have a free stream passing over the plate. You have correctly pointed out that there will be an region of elevated pressure in front of the plate. However, as the air in front is redirected and moves out over the edge of the plate, the free stream influence will tend to bend it back toward the free stream direction and will tend to drag the air behind the plate along with it. Once this process reaches steady state (in a time-averaged sense), this will tend to set up a pair of recirculating regions behind the plate as shown here:

That is the time-averaged flow, as the actual flow for any realistically large Reynolds number will be unsteady and what you will find is the development of a Kármán vortex sheet behind the plate, which will look something like this:

In each of these cases (time-averaged and time-resolved) the situation is identical to what would happen with a flat plate moving through stationary air.

 

[A.T.]

Why do you think the air behind is not affected by the stream coming from the fan?

He is considering a piston fitted in a cylinder there. No free stream at all. See post #21.

 

[boneh3ad]

He is considering a piston fitted in a cylinder there. No free stream at all. See post #21.

Ah, I missed that. Still, I think OP likely suggested that situation without realizing that it invalidates the original question since there is no one actually suggesting that those two cases would be identical with a wind tunnel completely blocked like that.

 

[leviterande]

Still, I think OP likely suggested that situation without realizing that it invalidates the original question

Dear boneh3ad, admittedly yes I know that this last example may look very different, but I wanted originally to consider this last example to show the same point I have in mind, namely the wake drag portion behind the plate.

Why do you think the air behind is not affected by the stream coming from the fan?

I understand that the air coming from the front and around the plate´s edges affects the air behind, induces vortices, and contributes to drag. I just thought that this edge rushing air from the front is only partly responsible for wake drag or drag on the back side of the plate. Besides this edge rushing air component of wake drag, I thought that there is another additional component of back side drag. I thought that the air mass originating from behind the plate played a roll and also partly responsible for the wake drag.

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?

 

[Chestermiller]

Dear boneh3ad, admittedly yes I know that this last example may look very different, but I wanted originally to consider this last example to show the same point I have in mind, namely the wake drag portion behind the plate.I understand that the air coming from the front and around the plate´s edges affects the air behind, induces vortices, and contributes to drag. I just thought that this edge rushing air from the front is only partly responsible for wake drag or drag on the back side of the plate. Besides this edge rushing air component of wake drag, I thought that there is another additional component of back side drag. I thought that the air mass originating from behind the plate played a roll and also partly responsible for the wake drag.

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?

No. Not in the case where the plate fills the tunnel.

The stationary plate Galilean version of this situation is where the plate is not moving, but the tunnel walls are sliding backwards. They are dragging air with them. So the flow is a combination of pressure flow and drag flow (drag by the walls). I see no one has realized this Galilean situation yet. Would you like me to draw the streamlines for this flow so that you can understand how the force on the back side of the plate comes about?

 

[boneh3ad]

Dear boneh3ad, admittedly yes I know that this last example may look very different, but I wanted originally to consider this last example to show the same point I have in mind, namely the wake drag portion behind the plate.I understand that the air coming from the front and around the plate´s edges affects the air behind, induces vortices, and contributes to drag. I just thought that this edge rushing air from the front is only partly responsible for wake drag or drag on the back side of the plate. Besides this edge rushing air component of wake drag, I thought that there is another additional component of back side drag. I thought that the air mass originating from behind the plate played a roll and also partly responsible for the wake drag.

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?

If you start a plate in motion in stature air until it reaches the same speed as in the wind tunnel case, then it will eventually reach the same state. Any effect of the mass of air behind the plate initially would fade as the air rushes around the edges and the wake develops.

Those vortices that develop will tend to throw any "excess" air out of that region anyway. Since they are rotating, there is a centripetal force involved which is provided by the pressure gradient, meaning the vortices will have low-pressure centers just like in the wind tunnel.

Regarding your example with a plate filling the tunnel, this cannot be used as an analogy. The walls would prevent any flow from going around the plate at all, which is not analogous to the moving plate in stationary air.

 

[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