An Explanation and Understanding of 1 Aerodynamic Lift by Triple-Deck Theory ∗ 2

5 An explanation of aerodynamic lift still is under controversial discus- 6 sion as can be seen, for example, in a recent published article in Scientiﬁc 7 American [1]. In contrast to the use of integral conservation laws we 8 here review an approach via the classical Kutta-Condition and its rela- 9 tion to boundary layer theory. Thereby we summarize known results for 10 viscous correction to the lift coeﬃcient for thin aerodynamic proﬁles and 11 try to remember the work on Triple-Deck Theory (TDT) or higher order 12 Boundary Layer theory. Connection to interactive boundary layer the- 13 ory, viscous/inviscid coupling as implemented to well-known engineering 14 code Xfoil is discussed. Finally we compare ﬁndings from tDT with 2D 15 numerical solutions of full Navier-Stokes equations (CFD)models. As a 16 conclusion, a clearer deﬁnition of terms like understanding and explana- 17 tion applied to the phenomenon of aerodynamic lift will be given. 18

sumptions resulting in equations for quantitative descriptions to be com-126 pared with measurements.  To remind to the basic concepts of Mechanics we may start by shortly referring 130 to Newton's 2 nd law for a point mass: A force (in N) relates to the temporal change of momentum p = m · v. A cause 132 and effect relationship may be established in both directions, meaning that a 133 force is a cause for a change in momentum, or a change in momentum may be 134 the cause for an inertial force. 135 Fluid mechanics as a continuum theory is formulated in terms of a velocity field v and expresses momentum change (mChange) and mass conservation (mCons): Here, and in the following we assume incompressible and sub-sonic flow. In- this equation may then be solved by the introduction of Green's function: where p harmonic (r) is a solution of the homogeneous pressure equation: 144 ∆p harmonic (r) = 0 .
We will come back to this in connection with formulating boundary conditions 145 for the pressure at the TE, se Eq (53) Contrarily to p f , the body-force density 146 usually is regarded as given from outside and in many cases does not have to be 147 included. (With an exception of the so-called Actuator Disk see [6].) 148 In the rest of the paper we proceed as follows: We review basic models of 149 lift in a more logical manner than they appeared historically, compare them appendix finally provides more technical details of the Triple-Deck-Theory. 153 We have to remark that our review mainly follows an approach in the spirit 154 of Landau [7] which from the beginning emphasis the role of a wake emerging 155 down stream of an aerodynamic profile. 156 Therefore it is rather different from that of McLean [5, 8,9] who emphasizes 157 the non-local pressure field as a direct result of the lift force [9] and its reciprocal 158 interaction with the velocity field as a key ingredient of a qualitative explanation 159 of lift and not as pessimistic as expressed in an already mentioned article by 160 Regis [1].
Here L is the lift force (per unit span), ρ the density of the fluid, w the inflow 180 velocity (far up-stream) and Γ the circulation, defined as A lift-curve slope of 2π -independent of all geometrical details -therefore is 209 predicted and the angle-of-attack (AOA) appears to be the most important Yates [26,24] combined Reynolds number and thickness corrections to γ E = 0.57722 being Euler's constant which shows a decrease of more than 10 246 % at t/c = 0.3 and RN around 10 5 from RN effects which -at least -is partly 247 gives further details.

249
Not included in all these discussions is the influence of the flow-state of 250 the boundary layer, whether it is laminar or turbulent. In our discussions we 251 assume that lift (in the linear part) is not influenced as strongly as drag. It 252 is well known that drag can be much higher when most parts of the boundary 253 layer are turbulent.

254
Another important phenomenon, flow separation, the starting point defined Separation usually limits c L (as measured) to values from 1.0 to about 2.0. We 257 will come back to that in section 4.5 as there is a close inter-dependency between 258 separation and some TAT scales . 259 We implicitly assume that the effect of separation can be approximately is predicted. Here, U e resp. U ∞ is the velocity at the edge of the boundary layer ( Here δ 1 and θ is the displacement thickness and momentum thickness at the 274 trailing edge, resp. the Navier-Stokes Equations to a still non-linear but much simpler ordinary 281 differential equation for an auxiliary function F with the stream function being 282 √ 2xF : together with boundary conditions F (0) = F (0) = 0 and F (s) → s as s → ∞.

284
Here with: λ = F (y = 0). With It must be noted, that today Eq. (16) is typically solved numerically to arbi- It follows [2]   However, it has to be added, that McLachlan [36] showed that this is mainly due to a fortunate cancellation of terms ∼ RN −1 , Blasius (solid line): Triple-Deck-Theory (dashed line): Value for Re = 10: Value for Re = 1000: Eq. (25) contains a new term ∼ Re −7/8 (note that the exponent is not -1 302 but -7/8) which contributes to more than 10 % to the drag and which will be 303 discussed in more detail below. for y=0 u=0 which is simply the no-slip condition. Within the wake (x > 0) 311 we will have u = 0. This different behavior at y = 0 for x < 0 and x > 0 is 312 the reason, that the wake exhibits a two-fold structure, separated by a curve 313 y ∼ x −1/3 , see Fig. 6. Unfortunately, close to x = 0 a singularity appears: Analogously to Eq. (16) the wake is composed of two boundary layers (inner 322 and outer wake) and therefore needs to be described in terms of two functions 323 H 0 , H 1 via: Using s = y/x 1/3 , it follows: Numerical integration leads to [2] λ 0 = 0.8789, λ 1 = −0.1496.

Boundary Layer Edge
Outer Wake Oncoming Boundary Layer

344
A sketch of the structure is visualized in Fig. 7.

345
Triple-Deck equations start with introducing appropriate scaled coordinates:

346
Main deck: Y = RN 1/2 y (34) Outer deck: and acts as a kind of a displacement function.

348
A characteristics set of equations can be derived [2]: together with asymptotics: In the appendices, see section (10) some more details of the mathematical prop-  . To scale A 1 in SI units it has to be multiplied by RN −3/8 . If RN=10 6 the scale then is 5.6 · 10 −3 . A 1 must obey the asymptotic limit µ0 λ0 X 1/3 as X → ∞ (red line). In addition A 1 (which is proportional to the pressure) is included which is ∈ C 0 (continuous) but / ∈ C 1 (continuous differentiable).

Sobey [2] provides a set of FORTRAN routines for solving this non-linear set
358 of integro-differential equations. Some sample results are presented in Fig. 8 359 and Fig. 9. As can be seen from the plots, all functions now are continuous at 360 x = 0 (trailing edge) but still are not ∈ C 1 (of continuous slope). For reasons of 361 comparison we added results from CFD in Fig. 9.

362
Pressure is shown in Fig. 9 together with two asymptotics p ∼ Singularity v → ∞ at the trailing edge (see Eq. 28) disappears, because [41] is finite for X → 0 ± . Unfortunately, streamlines close to the TE are not ∈   Figure 10: Same as Fig. 9, but enlarged. The apparently out-liner at X ≈ −0.3 might be due to inaccurate geometric modeling of TE. ment of prediction of finite length flat-plate drag coefficient. We will present 371 these findings in more detail in section 6. of. It is important to note that the inviscid solution is assumed to be in accordance with the most general inviscid flow around a 2D body.
Demand unique pressure at Y=0 : Eq. (53) may be regarded as some kind of a weaker Kutta-Joukovsky-condition  As a solver we use ANSYS-FLUENT V18 and ICEM/CFD for mesh preparation.

429
To have a more quantitative comparison of TDT we compared highly sensitive drag data either calculated from the above mentioned CFD-model or from Imai: as shown in table 1. Unfortunately, the contribution of the next-to-BL term c D Imai c D Dean 1.23 · 10 5 3.9 · 10 −3 3.04 3.81 · 10 −3 3.82 · 10 −3 1.0 · 10 4 1.43 · 10 −2 3.20 1.34 · 10 −2 1.35 · 10 −2 1.0 · 10 2 1.92 · 10 −1 3.31 1.14 · 10 −1 1.56 · 10 −1 Therefore, the wake with its continuous pressure in y-direction enforces an equal immediately after the trailing edge, TDT therefore will predict a reverse, that 467 is the pressure in the wake for y > 0 will be higher than the pressure in the 468 wake for y < 0 and so stabilize and maintain the flow leaving the trailing edge 469 tangentially.

470
The following list is intended to summarize this logical sequence of arguments Mass conservation then reads (as it must) 517 U 1,X + V 1,X = 0 (68) Writing down the momentum-equation [53] gives a set of two linear but coupled 518 pDEQs: maindeck: Here A 1 (X) as already introduced in Eq. (38) appears as an integration constant 522 of the equations of the middle deck: Namely 524 • Introduce Fourier transform with regard to X 529 P (ω) = 1 √ 2π P (X)e iωX dX (80) • and finally perform Fourier Inversion to receive at: A 1 already introduced in Eq. (40).

532
It is interesting to note that the above derived equation for the pressure dis- together with BoCos: (85) Using an Ansatz an Airy type of DEQ (f + Y · f = 0) follows with solution: