Dead-Downwind Faster Than The Wind (DDWFTTW) Analysis

Transcription

1 Dead-Downwind Faster Than The Wind (DDWFTTW) Analysis Mark Drela Jan 09 Nomenclature boat speed W wind speed F t drag force on water turbine thrust force on air prop P t shaft power of water turbine P p shaft power of air prop air-prop disk area (= πrp ) ρ η t η g η p η i η v air density water turbine efficiency gearing/transmission efficiency air prop total efficiency air prop induced efficiency air prop profile efficiency air prop swirl efficiency W wind speed air prop W P p Ω p R p boat speed water turbine apparent speeds P t F t elocities The figure above shows a boat moving with water-speed, in the same direction as a slower wind speed W. The water turbine therefore sees a water velocity of, while the air prop sees an air velocity of W, both opposite the boat motion. From the definition of turbine and propeller efficiencies: From the definition of the gear/transmission efficiency: Substituting for P p and P t above gives P t = F t η t () P p = ( W)/η p () P p = P t η g () ( W)/η p = F t η t η g () or = F t W η t η g η p (5) The net thrust available for overcoming the total vehicle water and air drag is therefore ( ) F net = F t = F t W η t η g η p which can be positive provided the following holds: (6) W η t η g η p > (requirement for DDWFTTW) (7)

2 Substituting F t = F net in equation (6) allows solving explicitly for the excess thrust F net. F net = { + [ ] } W η t η g η p (8) Air Prop Efficiency Breakdown The relation (8) above is not useful for initial-design estimation, since W and η p are both close to zero near the static-thrust condition, and their ratio is crucial. To resolve this problem, the air prop efficiency is broken down into a viscous (profile-drag) efficiency η v, and an inviscid (or induced) efficiency η i taken from actuator-disk theory. The latter is modified by including a swirl efficiency which accounts for non-axial velocities in the slipstream. η p = η i η v (9) η i = ( ) / (0) + + ρ( W) The F net relation (8) then takes the following equivalent form. η t η g η v F net = + ( ( W) + ( W) + F ) / p ρ () The very uncertain η p has now been eliminated, and the four remaining efficiencies can be realistically estimated a priori. Conservative values might be the following () η v 0.90 () η g 0.90 () η t 0.70 (5) The net thrust can now be quickly estimated as a function of the air prop thrust and the few other parameters, which is particularly useful for preliminary design of a DDWFTTW vehicle. Highly-Loaded Air Prop Limit When the vehicle is crossing the DDWFTTW threshold, the air prop s relative airspeed W is near zero, which means it s operating in hover mode. Near this important condition the diskloading term /ρ dominates the W terms in relation (), which can then be simplified as follows. ( ) / πρ F net = + R p η t η g η v (6) This is simpler and somewhat conservative, so it may be useful for preliminary sizing.

3 Dimensional Analysis For optimization purposes, it s useful to introduce the following dimensionless parameters which characterize the operation of any DDW machine. Excess-thrust ratio: F = F net (7) Apparent velocity ratio: Z = W (8) Modified air prop thrust coefficient: C T = (9) ρ The modified thrust coefficient is normalized with the vehicle speed, rather than the more conventional prop tip speed Ω p R p. Ultimately these are related through the turbine s advance ratio and the transmission gearing, although these details can be worked out later and do not need to be considered at this stage. With the definitions above, the net-thrust equation () can be put into the following dimensionless form: η t η g η v F = + ( ) / (0) Net Thrust-Drag Balance Z + Z + C T The retarding drag force associated with supporting the weight W of the vehicle can be characterized by a resistance coefficient C r. For a wheeled wehicle this C r would be the rolling-resistance coefficient, and for a hydrofoil boat this would be the drag/lift ratio. For a buoyancy hull, C r would be some function of the Froude and Reynolds numbers, whose details are not considered here. Adding on the air resistance, quantified by the air drag area CDA, gives the net thrust-drag balance as follows. F net = WC r + ρ ( W) CDA () F = W C r + Z C T CDA () Using this to replace F in (0) we get an implicit relation for Z. W C r + Z C T CDA = + Z + η t η g η v ( Z + C T ) / () This can be numerically solved for Z if all the other parameters are given. Once Z is known, the corresponding vehicle/wind speed ratio can be determined. W = Z ()

4 Numerical Investigation Equation () together with () defines the vehicle/wind speed ratio as a function of the following parameters. W = f (C T, η net,, C r, CDA ) (5) where η net = η t η g η v (6) CDA C r = W C r (7) = CDA (8) With a reasonable constraint on the vehicle length, which is needed to balance the moment of the high thrust line, the ratio W/ cannot be made much smaller than.0 or so. Hence C r is comparable to C r itself. The drag of the prop tower sets a lower limit on CDA. It is useful to now examine the sensitivity of /W to these parameters. The plot below shows /W versus C T, which can be strongly controlled by varying the air-prop diameter, for four values of η net, which depends mainly on the water-turbine and transmission efficiencies. The assumed values for the other parameters might be typical for a well-streamlined water vehicle with a good low-drag hull. = 0.95 C r = 0.5 CDA = 0.0 Given that η t = 0.7 is not easy for a small vehicle, reaching even η net = 0. might be challenging. Hence, achieving the DDWFTTW condition /W > would be quite difficult, but possibly doable with careful component design and matching. ehicle/wind speed ratio for eta_swirl = 0.95, Cr = 0.5, CDA = / W eta_net = 0.8 eta_net = 0.6 eta_net = 0. eta_net = CT Wheeled ehicles All the above definitions and equations easily apply to a wheeled ground vehicle, where the water

5 turbine is replaced by the wheels driven by the ground. If the wheel slip is negligible, then we have η t =. Also, C r now becomes the conventional rolling-resistance coefficient, which is dramatically smaller than the C r achievable by a hull. All other quantities should remain roughly the same. The plot below shows /W versus C T and η net, with the following assumed remaining parameters. = 0.95 C r = 0.0 CDA = 0.0 Since for this case η t, achieving η net = 0.6 or more is realistic. This confirms that the DDWFTTW condition /W > is achievable with a wheeled vehicle without too much difficulty. ehicle/wind speed ratio for eta_swirl = 0.95, Cr = 0.0, CDA = eta_net = 0.8 / W eta_net = 0.6 eta_net = 0. eta_net = CT 5