AVIATION ISSN: 1648-7788 / eISSN: 1822-4180 2019 Volume 23 Issue 1: 23–30 https://doi.org/10.3846/aviation.2019.10301
BALANCING AN AIRCRAFT WITH SYMMETRICALLY DEFLECTED SPLIT ELEVATOR AND RUDDER DURING SHORT LANDING RUN
Mohammed BA ZUHAIR*
Department of Aerohydrodynamics, Faculty of Aviation, Land Transport and Energetics, Kazan National Research Technical University – KAI, Republic of Tatarstan, 420097 Kazan, Russia
Received 11 July 2018; Accepted 23 April 2019
Abstract. This article investigates methods for balancing aircraft during short straight-line landing run realized by employ- ing split rudder and elevator as air-brakes after touchdown. For standard atmospheric and runway conditions, directional and longitudinal balance equations for aircraft of conventional configuration such as Il-86 are presented. Methods depend on operational and mechanical approaches, where the first requires manual or automatic trim of shortly peaking small pitching, yawing, and rolling moments using dynamic forces while the second suggest some re-design of elevator and rud- der control channels to limit deflection angles. The paper describes in detail each method disadvantages and suggests the adoption of automatic operational approach due to less required system modifications and piloting skills. Keywords: air-brake, short landing run, take-off and landing performance, split elevator and rudder, two sectioned rudder, two sectioned elevator.
Introduction American and Russian Space Shuttles were equipped with Improvement of take-off and landing characteristics of vertically unfolded split-rudders (NASA, 2007) function- wide-body transport aircraft remains one of the prior- ing as air-brakes during landing approach and run. Never- itized tasks whose innovative solution will increase profit- theless, large-scale adoption of the posterior/tail-mounted ability per flight. In general, particular interest is dedicated air-brakes and foldable split-rudders as supplementary to shorten take-off and landing phases for achieving less air-brakes have proven to be impractical for aircraft of fuel burn and noise as well as preserving urban resources conventional configuration because of control system by minimizing runway lengths and airport areas. complication and tail section overweighting. In addition, Short landing can be performed by implementing fuselage-mounted dorsal air-brakes require an unavailable three methods: aerodynamic, mechanical, and thrust vec- installation volume and lead to increase fuselage drag dur- toring. Mechanical method utilizes landing gear braking ing cruise flight (Mertol, 2008). system to utilize friction for aircraft deceleration, while Multi-functionalization of the existing pitch and yaw thrust vectoring is using thrust reversers for some back- sectioned control surfaces of wide-body long-haul aircraft ward reaction force generation. Aerodynamic method such as A380-800, B747-400, B747-8i, and Il96-300 etc. to as the most widespread is based on the full deployment execute air-brake function can be an alternative promising of high-lift devices, namely, air-brakes and lift-dumpers solution for shortening landing run especially consider- in addition to various types of conventional and uncon- ing the current availability of its required system compo- ventional spoilers (Mertol, 2008). Besides, posterior/tail- nents, which will overcome the aforementioned disadvan- mounted and fuselage-mounted dorsal air-brakes were tages. Therefore, some future aircraft concepts similar to applied, for instance, in Buccaneer and passenger aircraft BWB (Liebeck, 2004; Ba Zuhair, 2018) are considering BAE-146 (Jung, 2012). The application of sectioned con- integration of split drag-rudders, split drag-elevons, and trol surfaces for drag increase during landing mode were split drag-flaps with the design of future aircraft control previously introduced and implemented. For example, systems. chord-wise split elevons designed for B-2 are used for In this context, a new aerodynamic method-based quick deceleration and trim (Jung, 2012). In addition, both application is being developed to achieve short landing
*Corresponding author. E-mail: [email protected]
Copyright © 2019 The Author(s). Published by VGTU Press
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unre- stricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 24 M. Ba Zuhair. Balancing an aircraft with symmetrically deflected split elevator and rudder during short landing run by symmetrical deflection of SR and SE sections up to (gas or liquid) flows inside and outside SOLIDWORKS 15–45o after touchdown, a procedure synchronized with models, as well as heat transfer to (from, between, in) the deployment of high-lift devices and thrust reversers. these models due to convection, radiation, and conduc- Preliminary computational analysis within the framework tion with a proved CFD technology based on solving of this research showed landing run shortening of 5.6% RANS equations (Dassualt-Systems, 2015). Structured when thrust reversers are on and 7.1% when the latter is mesh statistics, simulation settings and validation results off. During abnormal weather conditions, every increase were detailed in (Bazuhair, 2018). Related initial condi- 6 of headwind speed by 1 m/s may contribute further short- tions include Vl set at 77.87 m/s and Re = 40.6×10 in ening of ~2.5% (Bazuhair, 2018). As a conclusion, section- stable atmospheric medium and standard runway surface ing rudder and elevator was introduced as effective inte- conditions recommended for numerical and experimental grated air-brakes. However, untrimmed pitching, rolling, tests (Federal Aviation Administration [FAA], 2018). Vali- and yawing moments were discovered after full deflection dation graph of CDα and CLα obtained by the simulation of the SR and SE sections primarily created by the differ- and Il-86 test flights shows sufficiently valid results around ent arm lengths of aerodynamic forces at each section of the interested αd =+3° governing post-touchdown landing the said control surfaces on the corresponding HS and VS. trajectory, see Figure 1. In this work, all effort is devoted to investigate opera- tional approaches capable of maintaining aircraft balance 2. Numerical analysis and discussion during a straight-line landing under standard weather conditions. Further analysis utilizes a mathematical model with geo- metrical and aerodynamic variables and constants indexed 1. CFD simulation and validation in accordance with the general form (j. k) explained as follows: “j” stands for tail part, i.e. elevator (symbolized Geometric and aerodynamic inputs for the numerical as – e.) or rudder (symbolized as – r.), while “k” refers study to solve the stability and trim mathematical equa- to the location of the attached control surface section to tions were imported from the computational analysis of the “j” with respect to the horizontal axis Ox of aircraft-body Russian aircraft Il-86. Firstly, a three-dimensional model coordinate system, where: right/left – r/l, root/tip – r/t, for of Il-86 with landing configuration was designed in and lower/upper – l/u. For example, the index “r.rr” be- SOLIDWORKS-2016 based on data from (Bekhtir, 1991), low Ar.rr is read as: “area of the right root sections of SR”, and further simulated using the built-in Flow Simulation see notions for additional examples. Calculations are run package with k-ɛ intensity and length turbulence model assuming these design parameters and operational con- code (Alyamovsky, 2012). Flow Simulation is a software siderations: fully integrated in SOLIDWORKS for computing fluid 1. Equal areas of SE and SR sections. Therefore, AAe.rk= e.lk = 0.25 A e , and AAr.u= r.l = 0.5 A r ; 2. ymmetrical deflection angles δ jk. of SE and SR sections. Accordingly, δr.l = −δ r.u , similarly δe.kr = −δ e.kt, δ=δ∈ where r.u e.kt 0 , 45 ; 3. Landing is performed on flat concrete runway at standard atmospheric conditions with functional engines, i.e. zero-sideslip angle. For analysis simplification, the investigated un- steady motion of aircraft is assumed to occur on three or more contact points with runway surface within a time- frame starting from touchdown to stop. Accordingly, air- craft may be observed as a rigid body with applied aero- dynamic and inertial forces including Lα, Dα , Wl , and T1j,2j with magnitude of nominal or reverse thrust mode values and relevant vector direction as well as Fn and Fr.k along with the relevant Nn and Nr.k. Additionally, aerody- namic forces Lj.k, Dj.k, and Zj.k from HS and VS, when SR and SE are symmetrically deflected are considered. Aero- dynamic forces Lal and Dal created by slightly deflectable ailerons may be introduced. Products of each said con- centrated and distributed forces multiplied by distances or force arms to the relevant local aerodynamic centers and GC form the static and control moments shown in Figure 1. Validation graph of the computed CDа and CLа Figure 2. against experimental data provided by (Bekhtir, 1991) Aviation, 2019, 23(1): 23–30 25
Figure 2. Forces influencing Il-86 during short landing run with symmetrically deflected SR and SE
Consequently, when neglecting deformation of tires In general, design of sectioned control surfaces pro- and shock-absorber struts at the moment of touchdown vides control system reliability enhancement and longitu- body-axis moment equations can be written as sums of dinal and directional controllability improvement at cruis- moment coefficients projected onOxy ,Oxz, Oyz as depict- ing speed (Bekhtir, 1991). Area of rudder and elevator ed in Figure 2: usually is split in half to ensure an equal distribution of the ∆ spanwise structural loads on each section. δr δδeal N ∑mmRx =∆ xxδ+re∆δmm+δ∑Ty ––mzх al w; (1) Let us consider system of Eqs (1)–(3) for aircraft of a Aq conventional landing configuration with respect to the de- δ δ scribed parameters and considerations in assumptions (1– n r δδ ∑mmRy=∑ Tx + m y δnr +∆ my δ + ee 3). Obviously, ∆=∆=mmxy0 and mm∑∑Ty= T = 0 δδ x ∆e δ−al δ − (2) because of flow and thrust symmetry. Safe landing mmye y al m yF∆ ; N run stipulates three-point contact with runway, thus m=++ mm m − m −− m m − = = ∑ Rz z∑ T zNn zN w.m zF n zF w.m ∑∑mmRx Rz 0 must be achieved. Only two degrees δδδ al j r e of motion freedom are permitted, i.e. forward along O x mzN− m zF + m z δal + mm z j+ z δr +∆ m z δ e . (3) g g w.s w.s of global coordinate system and yaw around Oy body-axis 26 M. Ba Zuhair. Balancing an aircraft with symmetrically deflected split elevator and rudder during short landing run
δr coordinate system. Accordingly, Eqs (1)–(3) will be rewrit- ∆>mx 0 one observes the opposite. However, for safe ten as follows: and stable landing run pressure on wheel brakes should δ δ ∆N not exceed the levels approved by the manufacturer the ∆mmr δ−al δ − z =0; (4) braking system. In addition, excessive wear of brakes and xxr alAq w one or more tires should be always avoided (FAA, 2018). δδδ nral Full deployment of air-brakes decelerates aircraft and ∑mmRy= y δn +∆ my δ r − m y δ al − m yF ; (5) ∆N significantly reduces effectiveness of its control surfaces including SR and SE as exemplified in Figure 5. Numerical mmz++∑ T m zN − m zN −− m zF m zF − m zN − δ n w.m n w.m w.s ∆ r δδδ solution of Eq. (5) reveals the trivial effect of mx below m+ mal δ + mmj j+r δ +∆ me δ =0. (6) zFw.s zal z zr z e 30m/s on the straightness of the landing run. Lowering Vl over landing timespan as demonstrated in Figure 5 and δr δ If δ=0 in Eqs (4–5), the created ∆m by arm r al x small ∆xr.k enable offsetting ∆mx using mechanical or length difference of Zr.k contributes an increase of the dy- aerodynamic approaches. namic loads on the right-side shock-absorber strut press- At speeds near Vl ailerons still effective, therefore, δ ing more on their tires, which is proportional to deflection al they can be set at a trimming angle δal. If mõ realizing angles δ of SR and SE as seen in Figures 3–4. j.k ∆=N 0 is known, then from Eq. (4) δal is: Such periodic loadings negatively affect service life of δ δ ∆m r tires. From Figure 2 (A) ∆m r is determined as: δ=x δ x al δ r . (8) m al δ δδ x ∆=r r.u δ −r.l δ mx CzVS kn VS r.u r.u y r.u CzVS knVS r.l r.l y r.l , (7) Analogically, using Eq. (5) one obtains:
AA δal δr where k= qq/ and nn= r.u , = r.l mmδ −∆ δ VS VS r.u AAr.l δ= óóal r n δ . (9) = n (Mkhitaryan et al., 2012). Also, nnr.u r.l according to my the above-mentioned assumption (1). Considering motion In general, airplane handbooks allow aileron imple- with αd and landing speed Vl the parameter kVS can be δ mentation for directional control in takeoff and landing ∆ δr Figure 3. Normal force increase on main landing Figure 5. Decrease of ∆mx for each δj.k during landing run gear right-side tires δ r δ Figure 4. Change of ∆mx as function of j.k Figure 6. Change of mRy as function of SR and SE deflection at δj.k Aviation, 2019, 23(1): 23–30 27 consequences. Moreover, transport aircraft are also al- this approach. In addition, complete elimination of the po- lowed to use rudder and aileron for landing stabilization tential extra physiological stress may remain unachieved δδrr during crosswinds (FAA, 2017, 2018). Note that the term since ∆mmxy ≠∆ as ∆xyrk.. ≠∆ rk. Moreover, the re- δal my in Eq. (9) also should count the effect of asymmet- quired re-design of travel limits of the SR with respect to ′ rical flow around ailerons due to Il-86 wing geometric δr.u decreases its maximum angles and complicates modi- twist, which approximately is –3° – (–4°) at low airspeeds fication process of the existing aircraft. Therefore, achiev- ′ (Bekhtir, 1991). When left aileron is deflected upwards, it ing δr.u is safer and more feasible using programmed or creates more aerodynamic forces than the right one that is manual differential deflection of SR sections to opposite δr deflected downwards. Mechanical approach suggests turn- unequal angles. Note that the problem of ∆mx charac- ing front wheel with δn to recover straight-forward land- terizes only tail sections with an odd number of fins. As ing run. Accordingly, on Table 1 trim angles δal and δn are for the cargo airplane An-225 with two fins and two SR, δ δ δ r listed. Note that trimming n is small, while al is effective deflection of both SR results in ∆=mxVS 0 , since δr.u and only within a short segment of runway length, where Vl > δr.l at each fin deflect oppositely. 30 m/s. On the other hand, from Eq. (6) one notices addi- tional pitching moments around Oz caused by deflecting Table 1. Required trim angles δal and δn SR and SE. Symmetrical deflection of SR always creates δ o o o some positive pitching moment m r > 0 , which increases δj δj.k = 15 δj.k = 30 δj.k = 45 z о о о by headwind: δal –4.9 –5.6 –6.6 δ δδ δ –1о –0.53о –0.73о r r.u r.l n mz = CxVS knVS r.u δ+ r.u y r.u CxVS knVS r.l δ r.l y r.l . (12) In scenarios where it is allowed, pilot may use differ- In fact, effects of SE symmetrical deflection on the to- ential braking to maintain directional control (FAA, 2018). tal pitching moment are complex. For sweptback HS, dif- ference between distances to GC from local aerodynamic This is implied by solving Eq. (5) with δ=al 0 resulting in δδ δ ∆=rn −δ ∆ r centers on deflected surfaces, i.e. xe.t and ye.t are a bit far- mmy yF∆ m y n, whereas for trimming my dif- δ N ∆ e ferential braking is expressed as follows (Buchkarev et. al, ther than xe.r and ye.r . This creates mz , which increases δe 1985): as χSt diverges. In result, there are two cases: ∆ Figure 7. Flow streamlines and interference behind the wing-tailing edge at Vl = 77.87 m/s viewed from: A – upper view on Oxz plane showing horizontal flow drifting; B – side view on Oxy plane showing vertical flow drifting ′ applying the same technique suggested in Eq. (11). Gener- Similarly, such a limited δe.t reduces SE air-braking δ e efficiency, which should be avoided. ally, ∆mz is given as: For aircraft with adjustable-incidence tailplane an al- δ δδ ∆=e δe.t +e.t − ternative method consisting in deflecting HS to some j mz kne.t e.t e.t( C yx e.t x e.t C e.t y e.t ) tr can be implemented. Such an approach trims the addi- δδ δ δ δ knδ+ Ce.r x Ce.r y . (13) e al r e.r e.r e.r( yx e.r e.r e.r e.r ) tional moments ∆mz , ∆mz , and ∆mz , while provid- ing good margin of longitudinal stability to prevent nose- = > wheel lift-off especially during relatively strong headwind. Here, nne.t e.r and kke.t e.r because of flow interference (Figure 7). However, this relation be- Using Eq. (6) Table 2 contains the required jtr for Il-86: comes kke.t< e.r in the second deflection option of SE δ ( δ = −δ ). Here, ∆ HL −mz0 −( x CG − xC Fc)( Lαα +∆ C L gr ) − j=tr jtr mz δδ δ (xxCm−) ∆HL − HL −∆ me δ −∆ mal δ −∆ mr δ CGF 2 La z0 ze z alz r . jtr mz (15) δ Figure 8. Change of mRz as function of j.k For tailless aircraft such as Concorde and Tu-144, as well as “flying wing” configuration such as B-2 flow drift- δe ing is zero. For Tu-144 χ=st 0 leading to ∆=mz 0 . Conclusions With the aim of addressing remarks outlined in a previ- ous numerical study concerning few landing run phase instabilities emerging after the deflection of tail part split control surfaces for shortening landing run of Il-86 air- craft this work detailed a number of operational and me- chanical approaches proposed to maintain aircraft balance during a straight-line landing run under standard weather δ and runway conditions. Mechanical approach is disadvan- Figure 9. Change of and r as function of δ mzHS mz j.k tageous and less competent because it requires re-design Aviation, 2019, 23(1): 23–30 29 of control surfaces attachments to include structural Liebeck, R. H. (2004). Design of the blended wing body subsonic limiters of deflection angles. In contrary, operational ap- transport. Journal of Aircraft, 41(1), 10-25. proaches comprise of less system architecture complex- https://doi.org/10.2514/1.9084 ity and suggest manual or automatic differential braking, Mertol, B. A. (2008). Patent application WIPO/2008/151760. Lifting wing with adjustable spoiler. nose wheel steering, aileron, and/or adjustable-incidence Mkhitaryan, A. M., et al. (2012). Dinamika poleta [Eng. trans. tailplane setting at specific trimming angles to offset the Flight Mechanics]. Moscow: EKOLIT publisher. untrimmed small pitching, yawing, and rolling moments NASA. (2007). Landing the space shuttle orbiter. NASA. Re- peaking directly after touchdown. The quick deceleration trieved from http://www.nasa.gov/pdf/167415main_Landin- during the landing run causes rapid attenuation of these gatKSC-08.pdf moments until reaching 30 m/s where they become insig- nificant. Up-to-date automatic high-lift and braking sys- tems incorporated in modern aircraft control programs Appendix provide the basis for successful modification of sectioned rudder and elevator control algorithms to qualify for safe Notations and effective application as air-brakes without overloading Variables and functions pilot or requiring his exceptional skills. Re – Reynold’s number; α Disclosure statement – angle of attack; CDα – drag coefficient at given α; The author declares that he has no financial, professional, C – lift coefficient at given α; or personal interests from other parties that relate to the Lα research described in this paper. αd – wing design angle of attack; Vl – landing airspeed; References Lα – lift at given α; Alyamovsky, A. A. (2012). SolidWorks Simulation, Kak reshat Dα – drag at given α; prakticheskie zadachi [SolidWorks Simulation. How to solve Wl – landing weight of aircraft; practical problems]. Saint Petersburg: Saint Petersburg Pub- T – thrust from pair engines 1; lisher. 1j Ba Zuhair, M. A. (2018). R.U. Patent No. 2,668,000. Moscow: T2j – thrust from pair engines 2; R.U. Patent and Trademark Office. Fn – friction force from nose wheel; Bazuhair, M. A. (2018). A technique for shortening landing run F – friction force from “k” nose wheel; distance of an aircraft by symmetrical deflection of split eleva- r.k tor and rudder. Russian Aeronautics, 61(2), 187-193. Nn – normal force from nose wheel; https://doi.org/10.3103/S106879981802006X Nr.k – normal force from “k” rare wheel; Bekhtir, V. P. (1991). Prakticheskaja ajerodinamika samoleta Il-86 [Il-86 aircraft performance]. Ulyanovsk: Center of Civil Avia- Lj.k – lift from slightly deflected at “j.k”; tion and Council for Mutual Economic Assistance. Dj.k – drag from slightly deflected at “j.k”; Buchkarev, A., et al. (1985). Aeromechanika samoleta [Aircraft Z – lateral force from slightly deflected at “j.k”; aeromechanics]. Moscow: Mashinostroenie. j.k Dassualt-Systems. (2015). Flow Simulation 2016 Online user’s Lal – lift slightly deflectable ailerons; guide. SOLIDWORKS-2016. Paris: Dassualt Systems. Re- Dal – drag from slightly deflectable ailerons; trieved from https://www.faa.gov/documentLibrary/media/ δ – deflection angle of SR sections; Advisory_Circular/AC_25-7D.pdf r δr Federal Aviation Administration. (2017). Approaches and land- ∆mx – partial derivative of rolling moment coefficient ings (Chapter 8). In Airplane Flying Handbook, FAA-H-8083- as function of δr; 3B. Retrieved from https://www.faa.gov/regulations_policies/ δ – deflection angle of SE sections; handbooks_manuals/aviation/airplane_handbook/media/10_ e δe afh_ch8.pdf ∆mx – partial derivative of rolling moment coefficient Federal Aviation Administration. (2018). Circular 25-7D- Flight as function of δe; Test Guide for Certification of Transport Category Airplanes. m∑Ty – rolling moment coefficient from the vertical Retrieved from https://www.faa.gov/documentLibrary/me- component of thrust by jet engines 1 and 2; dia/Advisory_Circular/AC_25-7D.pdf Federal Aviation Administration. (1997). Rules and Regulations: δal – angle of aileron deflection; δ Sino Swearingen Model SJ30–2 Airplane. Retrieved from al mx – rolling moment coefficient, when ailerons are https://www.govinfo.gov/content/pkg/FR-1997-10-31/pdf/97- deflected by δ ; 28937.pdf al Jung, U. S. (2012). Alternative air brake concepts for transport ΔN – change of normal forces on the main landing aircraft steep approach (PhD Thesis). Munich Technical Uni- gear right or left wheels due to rolling moment; versity, Munich, Germany. zw – distance between the main landing gear wheel tracks zw; 30 M. Ba Zuhair. Balancing an aircraft with symmetrically deflected split elevator and rudder during short landing run zw – relative distance between the main landing gear qVS – dynamic pressure around VS; wheel tracks; nr.k – relative effectiveness coefficients of each “k” SR d – wingspan; section; q – flow dynamic pressure; μ – friction coefficient; δr.u A – reference area of the wing; CxVS – derivative term of VS drag at δr.u > 0; δ m – yawing moment coefficient from the horizontal C r.l – derivative term of VS drag at δ > 0; ∑Tx xVS r.l component of thrust by engines 1 and 2; xe.t , xe.r – horizontal distance to local aerodynamic δn – nose wheel rotation angle; center of elevator tip “.t” or root “.r” sections; δn my – yawing moment coefficient of nose wheel; ye.t , ye.r – vertical distance to local aerodynamic cent- er of elevator tip “.t” or root “.r” sections; δr – deflection angle of rudder; ∆x – difference between horizontal distances from δe – deflection angle of elevator; r.k δ GC to local aerodynamic center of rudder tip “.t” or ∆m r – partial derivative of yawing moment coefficient y root “.r” sections; as function of δr; δ e χSt – stabilizer sweepback angle; ∆my – partial derivative of yawing moment coefficient δ as function of δ ; e.t e CLe.t – derivative term of SE lift coefficient at given δe.t ; δal mó – yawing moment coefficient, when ailerons are de- δ C e.r – derivative term of SE lift coefficient at given δ ; flected by δal; Le.r e.r δ m – yawing moment coefficient due to ΔN; e.t δ yF∆N CDe.t – derivative term of SE drag coefficient at given e.t ; m pitching moment coefficient of the steady-state δ z – e.r δ trimmed flight; CDe.r – derivative term of SE drag coefficient at given e.r ; HL m∑T – pitching moment coefficient from thrust; mz0 – pitching moment coefficient, when m – pitching moment coefficient from normal force C =δ =j=0 ; zNn La j. k created by nose wheels; x – relative center-of-gravity position; m – pitching moment coefficient from normal CG zNw.m force created by middle wheels; xFc – zero-α pitching moment coefficient from engines; m – pitching moment coefficient from normal zNw.s ∆C – incremental lift coefficient caused by ground force created by side (left and right) wheels; Lαgr effect; m – pitching moment coefficient from friction force zFn x – relative center of incremental lift caused by full created by nose wheels; F2 deployment of high-lift devices, or so-called “second” m – pitching moment coefficient from friction zFw.m focus; force created by middle wheels; HL ∆CLa – incremental lift coefficient caused by full de- m – pitching moment coefficient from friction zFw.s ployment of high-lift devices; force created by side (left and right) wheels; j – deflection angle of adjustable-incidence tailplane; δal tr mz – pitching moment coefficients, when ailerons are j m tr – pitching moment coefficient at j . deflected by δal; z tr j – incidence angle of HS; j Abbreviations mz – pitching moment coefficient influenced by inci- dence angle of HS (j); HS – horizontal stabilizer; δr mz – pitching moment coefficient, when δr > 0; VS – vertical stabilizer; δ e CFD – computational fluid dynamics; ∆mz – partial derivative of pitching moment coeffi- RANS – Reynolds Averaged Navier-Stokes; cient as a function of δe; δ SR – sections of rudder; C r.u – derivative term of VS lateral force at setting an- zVS SE – sections of elevator; gle δ ; r.u GC – center of gravity; δr.l CzVS – derivative term of VS lateral force at setting an- AHLCS – automatic high-lift control system; δ gle r.l ; MAC – main aerodynamic chord. kVS – coefficient of the flow deceleration around VS;