But you've gave the belt velocity as 10m/s in that post and the "flow" as being -6m/s... so according to your numbers, they're moving in opposite directions.
Can I assume that you left out the negative sign in front of the 10m/s by accident? And also left out the negative signs in front to your answers by accident too?
I think it is best not to assume anything, Brian-M...
You can't just leave out the negative signs on half your negative numbers and still expect your math to make any sense to anyone else.
Mmm...
Here, let's start again with a wind-speed cart.
On The Road
Measured from the ground - Beside the road.
Boundary layer (at height X) is 4m/s slower than the wind.
Wind = 10m/s
Cart = Wind = 10m/s
Boundary Layer = Wind - 4m/s = 6m/s
Road = 0
OK, so far, so good.
Now make the "wind" bit zero. The flow moves forward at the nominal 6m/s.
To measure from the cart's perspective (or wind's perspective, which is the same thing as they are moving at the same velocity), subtract the cart's velocity (10m/s) from everything.
Wind = 10m/s - 10m/s = 0
Cart = 10m/s - 10m/s = 0
Boundary Layer = 6m/s -10m/s = -4m/s
Road = 0 - 10m/s = -10m/s
Instead, add the wind.
Wind = 10m/s
Boundary = 6m/s.
The cart will be at 10m/s.
You can say 4 things. I think
(1)The Wind is faster than the boundary by 4m/s
(2)The Boundary is slower than the wind by 4m/s
(3)The Boundary is going backwards by 4m/s w.r.t the cart
(4) The difference ( Wind - Boundary) is 4m/s
Now we try it on the treadmill
On The Treadmill
Measured from the ground - beside the treadmill
The boundary layer (at height Y) is still 4m/s slower than the wind.
Wind = 0
Cart = 0
Boundary Layer = Wind - 4m/s = -4m/s
Belt = -10m/s
No. The belt speed
is the windspeed so, that is -10m/s. The flow is 4m/s,
slower, and so
-6m/s. As you would expect, 6m/s in the opposite direction. I suppose if you can "make the belt speed zero", the boundary will move along the belt at 6m/s.
It is exactly like the wind, but going the other way, its profile is the same "shape", but the fastest bit is actually at the road, going to zero at the wind. So it must be -6m/s.
However, the profile is not linear, so 6m/s will not be at the same depth in each case. Closer to the wind in the real case, but closer to the belt on the treadmill.
Wait a sec... That's EXACTLY the same as on the road, from the cart's perspective!
Let's look at it from the belt's perspective by subtracting belt-speed from everything.
Wind = 0 - (-10m/s) = 10m/s
Cart = 0 - (-10m/s) = 10m/s
Boundary Layer = -4m/s - (-10m/s) - 6m/s
Belt = -10m/s - (-10m/s) = 0
Exactly the same as we start with on the road!
There is no difference.
Not if the answer at windspeed is -6m/s
This is a trap I think. The paddle averages the forces if you will, so you can just assign a nominal value to the depth of the flow.
You are still ignoring the fact that the observer-generated wind and boundary flow pass that observer, up the belt, whereas the belt-boundary travels in the opposite direction. The paddle detects this too.
ETA:I imagine that the belt boundary is a thin layer of fog. Sit on the belt, but ignore the wind from behind. Part of you is travelling with the belt, but part is at zero wind. So for the -6m/s wind, you should see a flow, 4m/s going up the belt, backwards to you. This is like "winspeed" for the cart in real wind, (Option 3) but w.r.t to you.
But, there will be a wind of 10m/s from the rear, that makes, 6m/s down, 10m/s up, which is the same as the 4m/s 'backwards' up the belt +4m/s. At windspeed it is 6m/s down the belt, or -6m/s. or -4m/s if you take your view. Again, the paddle shows this.