Drag Forces in simple sweep theory...

[pilotmm]

Hello Folks

I want to calculate the sectional drag force vectors for a swept wing. According to simple sweep theory for lift one ignores the span-wise component of the flow, then in the reference frame perpendicular to the wing one calculates the AOA, look up Cl and voila you have your lift force. This all makes sense and I am quite comfortable with this...

However questions arise when calculating drag. If one calculates drag in this same swept wing reference frame (i.e. the drag vector is perpendicular to the wing) there would be a lateral component to the wing force. This component would be perpendicular to flow which does not make sense: It is basically lift turned side-ways. Considering the reaction in the flow from such a force, this would lead to a circulation about the vertical axis. (Thinking about Helmholtz laws of vorticity I find it hard to imagine what the vorticity system would look like in the wake from this vertical circulation). So I am not convinced simple sweep theory applies when calculating drag. I could understand a small lateral force from the pressure drag (due to rotated surface normal vectors), but not from the viscous drag (especially for sweep back, where the shear stress would point slightly outwards).

None of my textbooks really answer this question. Reading all the information easily accessible via google strangely there is much discussion about reducing wave drag yet nothing on actually calculating drag. I am hoping there is an expert on this forum that can answer this:

Please tell me given sectional drag coefficients and a the blades flow field how one would calculate the drag vectors?

I have two solutions:

1) Given aggregate Cd coefficients (pressure and viscous components combined), I would take the AOA in either the perpendicular frame or in the stream-wise frame frame (please advise what is correct), look up Cd and apply the stream-wise q and consider this force pointing in the stream-wise direction.

2) Given a pressure and viscous component of Cd I would calculate the pressure drag in the same way I calculate the lift, (using the perpendicular q and AOA) and consider this drag pointing in the direction of the projected wind vector (perpendicular to the wing, with small lateral component that I could accept). Then for the viscous component I would calculate according to solution 1.

Thanks a lot for your help.

Take Care
Mike

PS Posting a reference would be very helpful...

 

[Sparweb]

...i.e. the drag vector is perpendicular to the wing

Huh? Drag is only useful when applied to the direction of motion of the aircraft.
Don't let the spanwise component of flow (even though you know it's there) confuse your effort to define the flow in the axis of motion. You can deal with spanwise flow (3D effect) after you sort out the 2D geometry of the wing.

I don't know what aerodynamics books you are using, but try some by these authors:
(in order of increasing analytical depth)
Ansersen, John D
Hoerner
Perkins & Hage
Keuthe & Chow

STF

 

[pilotmm]

Thanks SparWeb for helping with this... I actually do not care too much about the lateral forces nor span-wise flow... I know realistically they are small enough to ignore at the design stage where simple sweep theory is relevant...

This is probably more of a geometry convention questions. So let me clarify...

For a swept wing there are two 2D planes, there is the plane perpendicular to the wing (it rotates about the vertical axis when the wing is swept (i.e. rotated about the same axis)) this is what I call the perpendicular plane/reference frame. So drag defined completely within this reference frame (no out-of-plane component) is perpendicular to the wing, this is what I was referring to as drag perpendicular to the wing.

Then there is a second plane that is not rotated and is defined by the flow and the vertical axis. The wing intersects with this plane at an oblique angle. Lets refer to this plane as the stream-wise plane. This plane does not rotate with the wing. This plane and the perpendicular one are the same only when the wing is not swept.

My understanding of Simple Sweep Theory (SST) is that the non-dimensional section lift coefficient is applied to the projected flow and geometry in the perpendicular plane. So the flow vector is projected on that plane, typically the sectional geometry of a wing is defined within that plane. So with the projected flow vector and geometry one has all the information they need to calculate lift. SST predicts that lift is reduced with swept wings because the project flow vector is less than the airspeed and hence the dynamic pressure is lower.

My questions starts when you extend this approach to calculate the drag force. In this case SST does not make complete sense, (maybe it does and I am just thoroughly confused...). So I wonder what is the conventional approach for calculating drag? You need 5 components: AOA, sectional coefficients, flow speed, chord and an assumed direction. I want to know what reference-frames/planes are these components defined within for the drag calculations.

So the first question is the selection of AOA and sectional coefficients. Since the sectional coefficients are defined in the perpendicular plane, then maybe it makes more sense to use the AOA defined in that same plane (which is slightly greater than the AOA in the stream-wise plane). Then when you consider the flow speed it makes more sense to use the true airspeed (that is driving the shear stress in the BL) which is in the stream-wise plane. Now this creates a geometric inconsistency, because the oblique profile as seen by the flow in this stream-wise plane is slightly different than the one in the perpendicular plane so maybe AOA should be based on the angle in this stream-wise plane?

Then I have more questions on the assumed direction of the drag force, typically this is the relative flow direction. If one assumes SST applies to drag and you ignore the span-wise component of flow, then that direction would be rotated (inwards in the case of backwards swept wings). This rotation does not make complete sense to me, the direction of viscous forces is driven by the flow and cannot be changed simply by rotating the wing in the same flow. So again I think the non-rotated stream-wise flow direction in the stream-wise plane should be used for this direction. This reinforces the geometric inconsistency.

What makes sense to me is if SST applies to lift, and lift is largely driven by pressure forces, then calculating the pressure drag by the same process (i.e. ignoring the span-wise flow for both dynamic pressure and direction) seems reasonable. Yet this does not make sense for viscous drag, there is makes more sense to use the geometry and AOA in the stream-wise plane. Without sectional coefficients for an oblique profile geometry than one has no choice but to use the perpendicular plane sectional coefficients, so maybe AOA for viscous drag from that plane makes more sense?

In the case where you do not have separate pressure and viscous drag coefficients. It makes more sense to use the flow speed and direction in the stream-wise plane. I still question the reference plane for the AOA convention (I lean towards stream-wise plane). I base this on the assumption that at low AOA viscous drag dominates.

I have two Andersen textbooks, maybe I just need to look harder but I did not find anything about simple sweep theory. I will track down those other books.

Thanks a lot for the help. Maybe I am a little too annoyingly fussy about geometric conventions. I do appreciate the help!
Mike