Low speed aerodynamics (second edition) part 2

CHAPTER 11 Two-Dimensional Numerical Solutions The principles of singular element based numerical solutions were introduced in Chapter and the first examples are provided in this chapter The following two-dimensional examples will have all the elements of more refined three-dimensional methods, but because of the simple two-dimensional geometry, the programming effort is substantially less Consequently, such methods can be developed in a short time for investigating improvements in larger codes and are also suitable for homework assignments and class demonstrations Based on the level of approximation of the singularity distribution, surface geometry, and type of boundary conditions, numerous computational methods can be constructed, some of which are presented in Table 11.1 We will not attempt to demonstrate all the possible combinations but will try to cover some of the most frequently used methods (denoted by the word “example” in Table 11.1), including discrete singular elements and constant-strength, linear, and quadratic elements (as an example for higher order singularity distributions) The different approaches in specifying the zero normal velocity boundary condition will be exercised and mainly the outer Neumann normal velocity and the internal Dirichlet boundary conditions will be used (and there are additional options, e.g., an internal Neumann condition) In terms of the surface geometry, for simplicity, only the flat panel element will be used here and in areas of high surface curvature the solution can be improved by using more panels In this chapter and in the following ones the primary concern is the simplicity of the explanation and the ease of constructing the numerical technique, while numerical efficiency considerations are secondary Consequently, the numerical economy of the methods presented can be improved (with some compromise in regard to the ease of code readability) Also, the methods are presented in their simplest form and each can be further developed to match the requirements of a particular problem Such improvement can be obtained by changing grid spacing and density, location of collocation points, or wake model, or altering the method of enforcing the boundary conditions and of enforcing the Kutta condition Also, it is recommended that one read this chapter sequentially since the first methods will be described with more details As the chapter evolves, some redundant details are omitted and the description may appear inadequate without reading the previous sections 11.1 Point Singularity Solutions The basic idea behind point singularity solutions is presented schematically in Fig 11.1 If an exact solution in a form of a continuous singularity distribution (e.g., a vortex distribution γ (x)) exists, then it can be divided into several finite segments (e.g., the segment x between x1 and x2 ) The local average strength of the element is then = x12 γ (x)d x and it can be placed at a point x0 within the interval x1 –x2 A discrete element numerical solution can be obtained by specifying N such unknown element strengths and then establishing N equations for their solution This can be done by specifying the boundary conditions at N 262 11.1 Point Singularity Solutions 263 Table 11.1 List of possible two-dimensional panel methods and of those tested in this chapter Boundary conditions Neumann (external) Singularity distribution Point Constant strength Linear strength Quadratic strength source doublet vortex source doublet vortex source doublet vortex source doublet vortex Dirichlet (internal) example example example example example example example Surface paneling flat/high-order flat example example flat flat flat flat flat flat example flat example example collocation points along the boundary Furthermore, when constructing the solution, some of the considerations mentioned in Section 9.3 (e.g., in regard to the Kutta condition and the wake) must be addressed As a first example for this very simple approach the lifting and thickness problems of thin airfoils are solved based on models (such as the lumped-vortex element) generated during examination of the analytical solutions in Chapter 11.1.1 Discrete Vortex Method The discrete vortex method presented here for solving the thin lifting airfoil problem is based on the lumped-vortex element and serves for solving numerically the integral equation (Eq (5.39)) presented in Chapter The advantage of the numerical approach is that the boundary conditions can be specified on the airfoil’s camber surface without a need for the small-disturbance approximation Also, two-dimensional interactions, such as those due to ground effect or multielement airfoils, can be studied with great ease Figure 11.1 Discretization of a continuous singularity distribution 264 11 / Two-Dimensional Numerical Solutions This method was introduced as an example in Section 9.8 and therefore its principles will be discussed here only briefly To establish the procedure for the numerical solution, the six steps presented in Section 9.7 are followed a Choice of Singularity Element For this discrete vortex method the lumped-vortex element is selected and its influence is given by Eq (9.31) (or Eqs (10.9) and (10.10)): u j = w 2πr 2j −1 x − xj z − zj (11.1) where r 2j = (x − x j )2 + (z − z j )2 Thus, the velocity at an arbitrary point (x, z) due to a vortex element of circulation j located at (x j , z j ) is given by Eq (11.1) This can be included in a subroutine, which will be called VOR2D: (u, w) = VOR2D( j , x, z, x j , z j ) (11.2) Such a subroutine is included in Program No in Appendix D b Discretization and Grid Generation At this phase the thin-airfoil camberline (Fig 11.2) is divided into N subpanels, which may be equal in length The N vortex points (x j , z j ) will be placed at the quarterchord point of each planar panel (Fig 11.2) The zero normal flow boundary condition can be fulfilled on the camberline at the three-quarter point of each panel These N collocation points (xi , z i ) and the corresponding N normal vectors ni along with the vortex points can be computed numerically or supplied as an input file Note that by discretizing the camberline as shown in Fig 11.2, we end up with only the panel edges remaining on the original camberline For convenience, the normal vector is evaluated at the actual camberline and the effect of this choice will be investigated at the end of this section Consequently, the normal vectors ni , pointing outward at each of these points, are approximated by using the Figure 11.2 Discrete vortex representation of the thin, lifting airfoil model 11.1 Point Singularity Solutions 265 Figure 11.3 Nomenclature used in defining the geometry of a point singularity based surface panel surface shape η(x), as shown in Fig 11.3: ni = (−dη/d x, 1) (dη/d x)2 + = (sin αi , cos αi ) (11.3) where the angle αi is defined as shown in Fig 11.3 Similarly the tangential vector ti is ti = (cos αi , − sin αi ) (11.3a) Since the lumped-vortex element is based on the Kutta condition, the last panel will inherently fulfill this requirement, and no additional specification of this condition is needed c Influence Coefficients The normal velocity component at each point on the camberline is a combination of the self-induced velocity and the free-stream velocity Therefore, the zero normal flow boundary condition can be presented as q·n=0 on solid surface Division of the velocity vector into the self-induced and free-stream components yields (u, w) · n + (U∞ , W∞ ) · n = on solid surface (11.4) where the first term is the velocity induced by the singularity distribution on itself (hence “self-induced part”) and the second term is the free-stream component Q∞ = (U∞ , W∞ ), as shown in Fig 11.2 The self-induced part can be represented by a combination of influence coefficients, while the free-stream contribution is known and will be transferred to the right-hand side of the boundary condition To establish the self-induced portion of the normal velocity, at each collocation point, consider the velocity induced by the jth vortex element at the first collocation point (in order to get the influence due to a unit strength j assume j = in Eq (11.2)): (u, w)1 j = VOR2D( j = 1, x1 , z , x j , z j ) (11.2a) The influence coefficient j is defined as the velocity component normal to the surface, due to a unit strength singularity element Consequently, the contribution of a unit strength singularity element j, at collocation point 1, is a1 j = (u, w)1 j · n1 (11.5) The induced normal velocity component qn1 , at point 1, due to all the elements is therefore qn1 = a11 + a12 Note that the strength of j + a13 + · · · + a1N is unknown at this point N 266 11 / Two-Dimensional Numerical Solutions Fulfillment of the boundary condition on the surface requires that at each collocation point the normal velocity component will vanish Specification of this condition (as in Eq (11.4)) for the first collocation point yields a11 + a12 + a13 + · · · + a1N N + (U∞ , W∞ ) · n1 = But, as mentioned earlier, the last term (free-stream component) is known and can be transferred to the right-hand side of the equation Consequently, the right hand side (RHS) is defined as RHSi = −(U∞ , W∞ ) · ni (11.6) Specifying the boundary condition for each of the N collocation points results in the following set of algebraic equations: ⎛ ⎞⎛ ⎞ ⎛ ⎞ a11 a12 a1N RHS1 ⎜ a21 a22 a2N ⎟ ⎜ ⎟ ⎜ RHS2 ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ a31 a32 a3N ⎟ ⎜ ⎟ ⎜ RHS3 ⎟ ⎜ ⎟⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ aN aN aN N N RHS N This influence coefficient calculation procedure can be accomplished by using two “DO loops” where the outer loop scans the collocation points and the inner scans the vortices DO i = 1, N DO j = 1, N C (collocation point loop) (vortex point loop) (u, w)i j = VOR2D( = 1.0, xi , z i , x j , z j ) j = (u, w)i j · ni CONTINUE END DO LOOP d Establish RHS Vector The right-hand side vector, which is the normal component of the free stream, can be computed within the outer loop of the previously described DO loops by using Eq (11.6), RHSi = −(U∞ , W∞ ) · ni where (U∞ , W∞ ) = Q ∞ (cos α, sin α) If we use the formulation of Eq (11.3) for the normal vector, the RHS becomes RHSi = −Q ∞ (cos α sin αi + sin α cos αi ) = −Q ∞ [sin(α + αi )] (11.6a) Note that α is the free-stream angle of attack (Fig 11.2) and αi is the ith panel inclination e Solve Linear Set of Equations The results of the previous computations can be summarized (for each collocation point i) as N j j=1 j = RHSi (11.7) 11.1 Point Singularity Solutions 267 Figure 11.4 Representation of a lifting flat plate by five discrete vortices For example consider the case of a flat plate (shown in Fig 11.4) where only five equal length elements ( c = c/5) were used Equation (11.7) for the five panels becomes ⎛ −1 ⎜ ⎜− ⎜ ⎜ ⎜ ⎜− π c⎜ ⎜ ⎜− ⎝ − 19 1 −1 1 − 13 −1 − 15 − 13 −1 − 17 − 15 − 13 ⎞ ⎟⎛ ⎟ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎟⎜ ⎟⎝ 1⎟ ⎠ −1 ⎞ ⎛ ⎞ ⎟ ⎜1⎟ 2⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎟ = −Q ∞ sin α ⎜1⎟ ⎝1⎠ 4⎠ This linear set of algebraic equations is diagonally dominant and can be solved by standard matrix methods Its solution is ⎛ ⎞ ⎛ ⎞ 2.46092 ⎜ 2⎟ ⎜1.09374⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = π cQ ∞ sin α ⎜0.70314⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 4⎠ ⎝0.46876⎠ 0.27344 and is shown schematically in Fig 11.5 Note that the total circulation is π cQ ∞ sin α, which is the exact result f Secondary Computations: Pressures, Loads, Velocities, Etc The resulting pressures and loads for this case can be computed by using the Kutta–Joukowski theorem for each panel j Thus the lift and pressure difference are L j = ρ Q∞ p j = ρ Q∞ where j j c (11.8) (11.9) c is the panel length The total lift and moment (about the leading edge) per unit 268 11 / Two-Dimensional Numerical Solutions Figure 11.5 Graphic representation of the computed vorticity distribution with a five-element discrete-vortex method span are obtained by summing the contribution of each element: N L= Lj (11.10) j=1 N M0 = − L j (x j cos α) (11.11) j=1 while the nondimensional coefficients become L Cl = (1/2)ρ Q 2∞ c M0 Cm = (1/2)ρ Q 2∞ c2 (11.12) (11.13) The following examples are presented to demonstrate possible applications of this method Example 1: Thin Airfoil with Parabolic Camber Consider the thin airfoil with parabolic camber of Section 5.4, where the camberline shape is x x η(x) = 1− c c For small values of < 0.1c the numerical results are close to the analytic results as shown in Fig 11.6 (here actually = 0.1 was used) This example can also be used to investigate the effect of the small-disturbance approximation (for the boundary conditions) on the pressure distribution, as shown by Figs 11.7 and 11.8 For the numerical solution the vortices were placed on the camberline where the boundary condition was satisfied For the analytical solution (and for the second numerical solution, aimed at simulating the analytical solution) the vortex distribution and the boundary condition were specified on the x axis The analytical pressure distribution can be obtained by substituting the coefficients A0 and A1 from Section 5.4 into Eqs (5.44a) and (5.48), which gives 2γ + cos θ =4 Cp = α+ sin θ Q∞ sin θ c 11.1 Point Singularity Solutions 269 Figure 11.6 Chordwise pressure difference for a thin airfoil with parabolic camber at zero angle of attack ( = 0.1) This can be rewritten in terms of the x coordinate by using Eq (5.45) (e.g., sin θ = 2[(x/c)(1 − x/c)]1/2 and cos θ = − 2x/c): Cp = c−x α + 32 x c x x 1− c c The effect of angle of attack is shown in Fig 11.8 where a fairly large angle (α = 10◦ ) is used Note the large suction peak at the leading edge, which is exaggerated by the thin airfoil solution In general, Figs 11.7 and 11.8 demonstrate that both thin-airfoil theory and the lumped-vortex panel method yield similar results A simple computer program using the principles of this section is presented in Appendix D, Program No Figure 11.7 Effect of small-disturbance boundary condition on the computed pressure difference on a thin parabolic camber airfoil (α = 0, = 0.1) 270 11 / Two-Dimensional Numerical Solutions Figure 11.8 Effect of small-disturbance boundary condition on the computed pressure difference on a thin parabolic camber airfoil (α = 10◦ , = 0.1) Example 2: Two-Element Airfoil The advantage of this numerical solution technique is that it is not limited to the restrictions of small-disturbance boundary conditions For example, a two-element airfoil with large deflection can be analyzed (and the results will have physical meaning when the actual flow is attached) Figure 11.9 shows the geometry of the two-element airfoil made up of circular arcs and the pressure difference distributions The interaction is shown by the plots of the close and separated elements (far from each other) When the elements are apart, the lift of the first element decreases while that of the second increases Example 3: Sensitivity to Grid After this first set of numerical examples, some possible pitfalls of the numerical approach can be observed (and hopefully avoided later) First note the method of paneling the gap region in the previous example of the two-element airfoil (Fig 11.10) If very few elements are used, then it is always advised to align the vortex points with vortex points and collocation points with collocation points We must remember that a numerical solution depends on the model and the grid (and hence is not unique) The convergence of a method can be tested by increasing the number of panels, which should result in a converging solution Therefore, it is always advisable to use smaller panels than the typical length of the geometry that we are modeling In the case of the two-element airfoil, the typical distance is the gap clearance, and (if possible with the more refined methods) paneling this area by elements of at least one-tenth the size of the gap is recommended 11.1 Point Singularity Solutions Figure 11.9 Effect of airfoil/flap proximity on their chordwise pressure difference spacing between the two elements 271 H is the vertical Another important observation can be made by trying to calculate the velocity induced by the five-element vortex representation of the flat plate of Fig 11.4 If the velocity survey is performed at z = 0.05c, then the wavy lines shown in Fig 11.11 are obtained This waviness will disappear at larger distances, and in any computation careful investigation is needed for the near and far field effects of a particular singular element distribution Figure 11.10 Method of paneling the gap region of a two-element airfoil (discrete-vortex model) D.3 Time-Dependent Programs 599 RZ=Z-Z1 R=SQRT(RX**2+RZ**2) IF(R.LT.0.001) GOTO V=0.5/PAY*GAMMA/R U=V*(RZ/R) W=V*(-RX/R) CONTINUE RETURN END C C PROGRAM No 16: UNSTEADY RECTANGULAR LIFTING SURFACE (VLM) C C THIS IS A 3-D LINEAR CODE FOR RECTANGULAR PLANFORMS (WITH GROUND EFFECT) IN UNSTEADY MOTION USING THE VORTEX LATTICE METHOD (BY JOE KATZ, 1975) DIMENSION ALF(5),SNO(5),CSO(5),ALAM(4),GAMA1J(5) DIMENSION QF(5,16,3),QC(4,13,3),BB(13),DLY(13) DIMENSION GAMA(4,13),DL(4,13),DP(4,13),DS(4,13),DLT(4,13),DD(4,13) DIMENSION A1(4,13),QW(50,14,3),VORTIC(50,13),UVW(50,14,3) DIMENSION QW1(50,14,3),VORT1(50,13),US(13) DIMENSION A(52,52),GAMA1(52),WW(52),DW(52),IP(52),ALAMDA(2) DIMENSION WTS(4,13),X15(50),Y15(50),Y16(50),Y17(50),Z15(50) COMMON VORTIC,QW,VORT1,QW1,QF,A1 COMMON IT,ALF,SNO,CSO,BB,QC,DS,ALAMDA,DXW COMMON/NO1/ SX,SZ,CS1,SN1,GAMA COMMON/NO2/ IB,JB,CH,SIGN COMMON/NO3/ IW C C C C C C C C C C C C C C C C C MODES OF OPERATION STEADY STATE : SET DT=DX/VT*IB*10, AND NSTEPS=5 SUDDEN ACCELERATION : SET DT=DX/VT/4 AND NSTEPS= UP TO 50 HEAVING OSCILLATIONS: BH=HEAVING AMPL OM= FREQ PITCH OSCILLATIONS : OMEGA=FREQUENCY, TETA=MOMENTARY ANGLE FOR COMPUTATIONAL ECONOMY THE PARAMETER IW MIGHT BE USED NOTE; INDUCED DRAG CALCULATION INCREASES COMPUTATION TIME AND CAN BE DISCONNECTED INPUT DATA IB=4 JB=13 NSTEPS=50 NW=5 DO 100 IPROG=1,IPROG1 NW - IS THE NUMBER OF (TIMEWISE) DEFORMING WAKE ELEMENTS IB = NUMBER OF CHORDWISE PANELS, JB = NO OF SPANWISE PANELS NSTEPS = NO OF TIME STEPS PAY=3.141592654 RO=1 BH=0.0 OM=0.0 VT=50.0 C=1 B=6.0 DX=C/IB DY=B/JB CH=10000.*C 600 C C C C C C C C C C C Appendix D / Sample Computer Programs C=CHORD; C=ROOT CHORD, B=SEMI SPAN, DX,DY=PANEL DIMENSIONS, CH=GROUND CLEARANCE, VT=FAR FIELD VELOCITY ALFA1=5.0 ALFAO=0.0 ALFA=(ALFA1+ALFAO)*PAY/180.0 DO I=1,IB ALF(I)=0 ALF(IB+1)=ALF(IB) ALF(IB+1) IS REQUIRED ONLY FOR QF(I,J,K) CALCULATION IN GEO ALAMDA(1)=90.*PAY/180 ALAMDA(2)=ALAMDA(1) ALAMDA(I) ARE SWEP BACK ANGLES (ALAMDA < 90, SWEEP BACKWARD) DT=DX/VT/4.0 T=-DT TIME IN SECONDS DXW=0.3*VT*DT DO J=1,JB BB(J)=DY CONSTANTS K=0 DO I=1,IB DO J=1,JB K=K+1 WW(K)=0.0 DLT(I,J)=0.0 VORTIC(I,J)=0.0 VORT1(I,J)=0.0 GAMA(I,J)=1 IS REQUIRED FOR INFLUENCE MATRIX CALCULATIONS GAMA(I,J)=1 CALCULATION OF COLLOCATION POINTS C 31 C C C C C C CALL GEO(B,C,S,AR,IB,JB,DX,DY,0.0,ALFA) GEO CALCULATES WING COLLOCATION POINTS QC,AND VORTEX TIPS QF WRITE(6,101) ALAM1=ALAMDA(1)*180./PAY ALAM2=ALAMDA(2)*180./PAY WRITE(6,102) ALFA1,ALAM1,B,C,ALAM2,S,AR,IB,JB,CH DO 31 I=1,IB ALL=ALF(I)*180./PAY WRITE(6,111) I,ALL DO I=1,JB,2 WRITE(6,105) I,BB(I) IB1=IB+1 JB1=JB+1 ============= PROGRAM START ============= DO 100 IT=1,NSTEPS C T=T+DT C C C PATH INFORMATION D.3 C C C C C C C C C C C C C C C C 10 C 11 C Time-Dependent Programs SX=-VT*T DSX=-VT CH1=CH IF(CH.GT.100.0) CH1=0.0 SZ=BH*SIN(OM*T)+CH1 DSZ=BH*OM*COS(OM*T) DSX=DSX/DT DSZ=DSZ/DT TETA=0.0 OMEGA=0.0 VT=-COS(TETA)*DSX-SIN(TETA)*DSZ SN1=SIN(TETA) CS1=COS(TETA) WT=SN1*DSX-CS1*DSZ DO I=1,IB SNO(I)=SIN(ALFA+ALF(I)) CSO(I)=COS(ALFA+ALF(I)) =========================== VORTEX WAKE SHEDDING POINTS =========================== DO J=1,JB1 QW(IT,J,1)=QF(IB1,J,1)*CS1-QF(IB1,J,3)*SN1+SX QW(IT,J,2)=QF(IB1,J,2) QW(IT,J,3)=QF(IB1,J,1)*SN1+QF(IB1,J,3)*CS1+SZ CONTINUE ======================== AERODYNAMIC CALCULATIONS ======================== INFLUENCE COEFFICIENTS CALCULATION K=0 DO 14 I=1,IB DO 14 J=1,JB SIGN=0.0 K=K+1 IF(IT.GT.1) GOTO 12 MATRIX COEFFICIENTS CALCULATION OCCURS ONLY ONCE FOR THE TIME-FIXED-GEOMETRY WING CALL WING(QC(I,J,1),QC(I,J,2),QC(I,J,3),GAMA,U,V,W) L=0 DO 10 I1=1,IB DO 10 J1=1,JB L=L+1 A(K,L) - IS THE NORMAL VELOCITY COMPONENT DUE TO A UNIT VORTEX LATTICE A(K,L)=A1(I1,J1) ADD INFLUENCE OF WING OTHER HALF PART CALL WING(QC(I,J,1),-QC(I,J,2),QC(I,J,3),GAMA,U,V,W) L=0 DO 11 I1=1,IB DO 11 J1=1,JB L=L+1 A(K,L)=A(K,L)+A1(I1,J1) IF(CH.GT.100.0) GOTO 12 ADD INFLUENCE OF MIRROR IMAGE SIGN=10.0 XX1=QC(I,J,1)*CS1-QC(I,J,3)*SN1+SX 601 602 C 12 C 121 122 C C 13 C C C C C C 14 C C C 15 Appendix D / Sample Computer Programs ZZ1=QC(I,J,1)*SN1+QC(I,J,3)*CS1+SZ XX2=(XX1-SX)*CS1+(-ZZ1-SZ)*SN1 ZZ2=-(XX1-SX)*SN1+(-ZZ1-SZ)*CS1 CALL WING(XX2,QC(I,J,2),ZZ2,GAMA,U,V,W) L=0 DO I1=1,IB DO J1=1,JB L=L+1 A(K,L)=A(K,L)+A1(I1,J1) ADD MIRROR IMAGE INFLUENCE OF WING'S OTHER HALF CALL WING(XX2,-QC(I,J,2),ZZ2,GAMA,U,V,W) L=0 DO I1=1,IB DO J1=1,JB L=L+1 A(K,L)=A(K,L)+A1(I1,J1) SIGN=0.0 CONTINUE IF(IT.EQ.1) GOTO 13 CALCULATE WAKE INFLUENCE XX1=QC(I,J,1)*CS1-QC(I,J,3)*SN1+SX ZZ1=QC(I,J,1)*SN1+QC(I,J,3)*CS1+SZ CALL WAKE(XX1,QC(I,J,2),ZZ1,IT,U,V,W) CALL WAKE(XX1,-QC(I,J,2),ZZ1,IT,U1,V1,W1) IF(CH.GT.100) GOTO 121 CALL WAKE(XX1,QC(I,J,2),-ZZ1,IT,U2,V2,W2) CALL WAKE(XX1,-QC(I,J,2),-ZZ1,IT,U3,V3,W3) GOTO 122 U2=0.0 U3=0.0 V2=0.0 V3=0.0 W2=0.0 W3=0.0 CONTINUE WAKE INDUCED VELOCITY IS GIVEN IN INERTIAL FRAME U=U+U1+U2+U3 W=W+W1-W2-W3 U11=U*CS1+W*SN1 W11=-U*SN1+W*CS1 WW(K) IS THE PREPENDICULAR COMPONENT OF WAKE INFLUENCE TO WING WW(K)=U11*SNO(I)+W11*CSO(I) CONTINUE CALCULATE WING GEOMETRICAL DOWNWASH DW(K)=-VT*SNO(I)+QC(I,J,1)*OMEGA-WT FOR GENERAL MOTION DW(K)=-VT*SIN(ALFA)+OMEGA*X WTS(I,J)=W11 W11 - IS POSITIVE SINCE THE LATEST UNSTEADY WAKE ELEMENT IS INCLUDED IN SUBROUTINE WING CONTINUE SOLUTION OF THE PROBLEM: K1=IB*JB DO 15 K=1,K1 GAMA1(K)=DW(K)-WW(K) IF(IT.GT.1) GOTO 16 DW(I)=WW(I)+A(I,J)*GAMA(I) D.3 C C 16 C C C C C 17 C C C C 162 171 C C C C C C 18 C 19 C 193 C C C C C Time-Dependent Programs FOR NONVARIABLE WING GEOMETRY (WITH TIME), MATRIX INVERSION IS DONE ONLY ONCE CALL DECOMP(K1,52,A,IP) CONTINUE CALL SOLVER(K1,52,A,GAMA1,IP) HERE * THE SAME ARRAY SIZE IS REQUIRED, AS SPECIFIED IN THE BEGINNING OF THE CODE WING VORTEX LATTICE LISTING K=0 DO 17 I=1,IB DO 17 J=1,JB K=K+1 GAMA(I,J)=GAMA1(K) WAKE SHEDDING DO 171 J=1,JB LATEST WAKE ELEMENTS LISTING VORTIC(IT,J)=GAMA(IB,J) VORTIC(IT+1,J)=0.0 CONTINUE =========================== WAKE ROLLUP CALCULATION =========================== IW=1 IF(IT.EQ.1) GOTO 193 IF(IT.GE.NW) IW=IT-NW+1 NW IS THE NUMBER OF (TIMEWISE) DEFORMING WAKE ELEMENTS I1=IT-1 JS1=0 JS2=0 DO 18 I=IW,I1 DO 18 J=1,JB1 CALL VELOCE(QW(I,J,1),QW(I,J,2),QW(I,J,3),U,V,W,IT,JS1,JS2) UVW(I,J,1)=U*DT UVW(I,J,2)=V*DT UVW(I,J,3)=W*DT CONTINUE DO 19 I=IW,I1 DO 19 J=1,JB1 QW(I,J,1)=QW(I,J,1)+UVW(I,J,1) QW(I,J,2)=QW(I,J,2)+UVW(I,J,2) QW(I,J,3)=QW(I,J,3)+UVW(I,J,3) CONTINUE CONTINUE ================== FORCES CALCULATION ================== FL=0 FD=0 FM=0 603 604 C C C 194 195 C Appendix D / Sample Computer Programs FG=0.0 QUE=0.5*RO*VT*VT DO 20 J=1,JB SIGMA=0 SIGMA1=0.0 DLY(J)=0 DO 20 I=1,IB IF(I.EQ.1) GAMAIJ=GAMA(I,J) IF(I.GT.1) GAMAIJ=GAMA(I,J)-GAMA(I-1,J) DXM=(QF(I,J,1)+QF(I,J+1,1))/2 DXM IS VORTEX DISTANCE FROM LEADING EDGE SIGMA1=(0.5*GAMAIJ+SIGMA)*DX SIGMA=GAMA(I,J) DFDT=(SIGMA1-DLT(I,J))/DT DFDT IS THE VELOCITY POTENTIAL TIME DERIVATIVE DLT(I,J)=SIGMA1 DL(I,J)=RO*(VT*GAMAIJ+DFDT)*BB(J)*CSO(I) INDUCED DRAG CALCULATION CALL WINGL(QC(I,J,1),QC(I,J,2),QC(I,J,3),GAMA,U1,V1,W1) CALL WINGL(QC(I,J,1),-QC(I,J,2),QC(I,J,3),GAMA,U2,V2,W2) IF(CH.GT.100.0) GOTO 194 XX1=QC(I,J,1)*CS1-QC(I,J,3)*SN1+SX ZZ1=QC(I,J,1)*SN1+QC(I,J,3)*CS1+SZ XX2=(XX1-SX)*CS1+(-ZZ1-SZ)*SN1 ZZ2=-(XX1-SX)*SN1+(-ZZ1-SZ)*CS1 CALL WINGL(XX2,QC(I,J,2),ZZ2,GAMA,U3,V3,W3) CALL WINGL(XX2,-QC(I,J,2),ZZ2,GAMA,U4,V4,W4) GOTO 195 W3=0 W4=0 W8=W1+W2-W3-W4 ADD INFLUENCE OF MIRROR IMAGE (GROUND) CTS=-(WTS(I,J)+W8)/VT DD1=RO*BB(J)*DFDT*SNO(I) DD2=RO*BB(J)*VT*GAMAIJ*CTS DD(I,J)=DD1+DD2 C 20 C C C C C C C C C DP(I,J)=DL(I,J)/DS(I,J)/QUE DLY(J)=DLY(J)+DL(I,J) FL=FL+DL(I,J) FD=FD+DD(I,J) FM=FM+DL(I,J)*DXM FG=FG+GAMAIJ*BB(J) CONTINUE CL=FL/(QUE*S) CD=FD/(QUE*S) CM=FM/(QUE*S*C) CLOO=2.*PAY*ALFA/(1.+2./AR) IF(ABS(CLOO).LT.1.E-20) CLOO=CL CLT=CL/CLOO CFG=FG/(0.5*VT*S)/CLOO ====== OUTPUT ====== PLACE PLOTTER OUTPUT HERE (e.g T,SX,SZ,CL,CD,CM) OTHER OUTPUT D.3 211 21 23 C C 100 C C C 101 102 103 104 105 106 107 108 109 110 111 112 113 C Time-Dependent Programs 605 WRITE(6,106) T,SX,SZ,VT,TETA,OMEGA WRITE(6,104) CL,FL,CM,CD,CLT,CFG I2=5 IF(IT.NE.I2) GOTO 100 WRITE(6,110) DO 21 J=1,JB DO 211 I=2,IB GAMA1J(I)=GAMA(I,J)-GAMA(I-1,J) DLYJ=DLY(J)/BB(J) WRITE(6,103) J,DLYJ,DP(1,J),DP(2,J),DP(3,J),DP(4,J),GAMA(1,J), 1GAMA1J(2),GAMA1J(3),GAMA1J(4) IF(IT.NE.I2) GOTO 100 WRITE(6,107) DO 23 I=1,IT WRITE(6,109) I,(VORTIC(I,K1),K1=1,13) DO 23 J=1,3 WRITE(6,108) J,(QW(I,K,J),K=1,14) CONTINUE END OF PROGRAM CONTINUE FORMATS FORMAT(1H ,/,20X,'WING LIFT DISTRIBUTION CALCULATION (WITH GROUND EFFECT)',/,20X,56('-')) FORMAT(1H ,/,10X,'ALFA:',F10.2,8X,'LAMDA(1) :',F10.2,8X,'B :', 1F10.2,8X,'C :',F13.2,/,33X, 2'LAMDA(2) :',F10.2,8X,'S :',F10.2,8X,'AR :',F13.2,/,33X, 3'IB :',I10,8X,'JB :',I10,8X,'L.E HEIGHT:', F6.2,/) FORMAT(1H ,I3,' I ',F9.3,' II ',4(F9.3,' I '),' I ',4(F9.3,' I ')) FORMAT(1H ,'CL=',F10.4,2X,'L=',F10.4,4X,'CM=',F10.4,3X,'CD=',F10.4 1,3X,'L/L(INF)=',F10.4,4X,'GAMA/GAMA(INF)=',F10.4,/) FORMAT(1H ,9X,'BB(',I3,')=',F10.4) FORMAT(1H ,/,' T=',F10.2,3X,'SX=',F10.2,3X,'SZ=',F10.2,3X,'VT=', 1F10.2,3X,'TETA= ',F10.2,6X,'OMEGA= ',F10.2) FORMAT(1H ,//,' WAKE ELEMENTS,',//) FORMAT(1H ,'QW(',I2,')=',22(F6.2)) FORMAT(1H ,' VORTIC(IT=',I3,')=',17(F6.3)) FORMAT(1H ,/,5X,'I DL',4X,'II',22X,'DCP',22X,'I I',25X, 1'GAMA',/,118('='),/,5X,'I',15X,'I= 1',11X,'2',11X,'3',11X, 2'4',5X,'I I',5X,'1',11X,'2',11X,'3',11X,'4',/,118('=')) FORMAT(1H ,9X,'ALF(',I2,')=',F10.4) FORMAT(1H ,'QF(I=',I2,',J,X.Y.Z)= ',15(F6.1)) FORMAT(1H ,110('=')) STOP END C C C C C C C SUBROUTINE VORTEX(X,Y,Z,X1,Y1,Z1,X2,Y2,Z2,GAMA,U,V,W) USE THIS SUBROUTINE FROM PROGRAM NO 13 SUBROUTINE WAKE(X,Y,Z,IT,U,V,W) DIMENSION VORTIC(50,13),QW(50,14,3) COMMON VORTIC,QW COMMON/NO2/ IB,JB,CH,SIGN COMMON/NO3/ IW CALCULATES SEMI WAKE INDUCED VELOCITY AT POINT (X,Y,Z) AT T=IT*DT, IN THE INERTIAL FRAME OF REFERENCE 606 Appendix D / Sample Computer Programs U=0 V=0 W=0 I1=IT-1 DO I=1,I1 DO J=1,JB VORTEK=VORTIC(I,J) CALL VORTEX(X,Y,Z,QW(I,J,1),QW(I,J,2),QW(I,J,3),QW(I+1,J,1), 1QW(I+1,J,2),QW(I+1,J,3),VORTEK,U1,V1,W1) CALL VORTEX(X,Y,Z,QW(I+1,J,1),QW(I+1,J,2),QW(I+1,J,3),QW(I+1,J+1,1 2),QW(I+1,J+1,2),QW(I+1,J+1,3),VORTEK,U2,V2,W2) CALL VORTEX(X,Y,Z,QW(I+1,J+1,1),QW(I+1,J+1,2),QW(I+1,J+1,3), 3QW(I,J+1,1),QW(I,J+1,2),QW(I,J+1,3),VORTEK,U3,V3,W3) CALL VORTEX(X,Y,Z,QW(I,J+1,1),QW(I,J+1,2),QW(I,J+1,3),QW(I,J,1), 4QW(I,J,2),QW(I,J,3),VORTEK,U4,V4,W4) U=U+U1+U2+U3+U4 V=V+V1+V2+V3+V4 W=W+W1+W2+W3+W4 CONTINUE RETURN END C C C C C C SUBROUTINE VELOCE(X,Y,Z,U,V,W,IT,JS1,JS2) DIMENSION GAMA(4,13) COMMON/NO1/ SX,SZ,CS1,SN1,GAMA COMMON/NO2/ IB,JB,CH,SIGN SUBROUTINE VELOCE CALCULATES INDUCED VELOCITIES DUE TO THE WING AND ITS WAKES IN A POINT (X,Y,Z) GIVEN IN THE INERTIAL FRAME OF REFERENCE X1=(X-SX)*CS1+(Z-SZ)*SN1 Y1=Y Z1=-(X-SX)*SN1+(Z-SZ)*CS1 CALL WAKE(X,Y,Z,IT,U1,V1,W1) CALL WAKE(X,-Y,Z,IT,U2,V2,W2) CALL WING(X1,Y1,Z1,GAMA,U3,V3,W3) CALL WING(X1,-Y1,Z1,GAMA,U4,V4,W4) U33=CS1*(U3+U4)-SN1*(W3+W4) W33=SN1*(U3+U4)+CS1*(W3+W4) INFLUENCE OF MIRROR IMAGE IF(CH.GT.100.0) GOTO X2=(X-SX)*CS1+(-Z-SZ)*SN1 Z2=-(X-SX)*SN1+(-Z-SZ)*CS1 CALL WAKE(X,Y,-Z,IT,U5,V5,W5) CALL WAKE(X,-Y,-Z,IT,U6,V6,W6) CALL WING(X2,Y1,Z2,GAMA,U7,V7,W7) CALL WING(X2,-Y1,Z2,GAMA,U8,V8,W8) U77=CS1*(U7+U8)-SN1*(W7+W8) W77=SN1*(U7+U8)+CS1*(W7+W8) GOTO CONTINUE U5=0.0 U6=0.0 U77=0.0 V5=0.0 V6=0.0 V7=0.0 V8=0.0 W5=0.0 D.3 C Time-Dependent Programs 607 W6=0.0 W77=0.0 CONTINUE VELOCITIES MEASURED IN INERTIAL FRAME U=U1+U2+U33+U5+U6+U77 V=V1-V2+V3-V4+V5-V6+V7-V8 W=W1+W2+W33-W5-W6-W77 RETURN END C SUBROUTINE WING(X,Y,Z,GAMA,U,V,W) DIMENSION GAMA(4,13),QF(5,16,3),A1(4,13),VORTIC(50,13),QW(50,14,3) DIMENSION ALF(5),SNO(5),CSO(5),VORT1(50,13),QW1(50,14,3) COMMON VORTIC,QW,VORT1,QW1,QF,A1 COMMON IT,ALF,SNO,CSO COMMON/NO2/ IB,JB,CH,SIGN C C C CALCULATES SEMI WING INDUCED VELOCITY AT A POINT (X,Y,Z) DUE TO WI VORTICITY DISTRIBUTION GAMA(I,J) IN A WING FIXED COORDINATE SYSTE U=0 V=0 W=0 DO I=1,IB DO J=1,JB C CALL VORTEX(X,Y,Z,QF(I,J,1),QF(I,J,2),QF(I,J,3),QF(I,J+1,1),QF(I,J 1+1,2),QF(I,J+1,3),GAMA(I,J),U1,V1,W1) CALL VORTEX(X,Y,Z,QF(I,J+1,1),QF(I,J+1,2),QF(I,J+1,3),QF(I+1,J+1,1 2),QF(I+1,J+1,2),QF(I+1,J+1,3),GAMA(I,J),U2,V2,W2) CALL VORTEX(X,Y,Z,QF(I+1,J+1,1),QF(I+1,J+1,2),QF(I+1,J+1,3), 3QF(I+1,J,1),QF(I+1,J,2),QF(I+1,J,3),GAMA(I,J),U3,V3,W3) CALL VORTEX(X,Y,Z,QF(I+1,J,1),QF(I+1,J,2),QF(I+1,J,3),QF(I,J,1), 4QF(I,J,2),QF(I,J,3),GAMA(I,J),U4,V4,W4) C U0=U1+U2+U3+U4 V0=V1+V2+V3+V4 W0=W1+W2+W3+W4 A1(I,J)=U0*SNO(I)+W0*CSO(I) IF(SIGN.GE.1.0) A1(I,J)=U0*SNO(I)-W0*CSO(I) U=U+U0 V=V+V0 W=W+W0 C CONTINUE RETURN END C SUBROUTINE WINGL(X,Y,Z,GAMA,U,V,W) DIMENSION GAMA(4,13),QF(5,16,3),A1(4,13),VORTIC(50,13),QW(50,14,3) DIMENSION ALF(5),SNO(5),CSO(5),VORT1(50,13),QW1(50,14,3) COMMON VORTIC,QW,VORT1,QW1,QF,A1 COMMON IT,ALF,SNO,CSO COMMON/NO2/ IB,JB,CH,SIGN C C C C C C CALCULATES INDUCED VELOCITY AT A POINT (X,Y,Z) DUE TO LONGITUDINAL VORTICITY DISTRIBUTION GAMAX(I,J) ONLY(SEMI-SPAN), IN A WING FIXED COORDINATE SYSTEM + (T.E UNSTEADY VORTEX) ** SERVES FOR INDUCED DRAG CALCULATION ONLY ** 608 Appendix D / Sample Computer Programs U=0 V=0 W=0 DO I=1,IB DO J=1,JB C CALL VORTEX(X,Y,Z,QF(I,J+1,1),QF(I,J+1,2),QF(I,J+1,3),QF(I+1,J+1,1 2),QF(I+1,J+1,2),QF(I+1,J+1,3),GAMA(I,J),U2,V2,W2) CALL VORTEX(X,Y,Z,QF(I+1,J,1),QF(I+1,J,2),QF(I+1,J,3),QF(I,J,1), 4QF(I,J,2),QF(I,J,3),GAMA(I,J),U4,V4,W4) C C C C U=U+U2+U4 V=V+V2+V4 W=W+W2+W4 CONTINUE ADD INFLUENCE OF LATEST UNSTEADY WAKE ELEMENT: I=IB DO J=1,JB CALL VORTEX(X,Y,Z,QF(I+1,J+1,1),QF(I+1,J+1,2),QF(I+1,J+1,3), 3QF(I+1,J,1),QF(I+1,J,2),QF(I+1,J,3),GAMA(I,J),U3,V3,W3) U=U+U3 V=V+V3 W=W+W3 CONTINUE RETURN END C SUBROUTINE GEO(B,C,S,AR,IB,JB,DX,DY,DGAP,ALFA) DIMENSION BB(13),ALF(5),SN(5),CS(5),SNO(5),CSO(5) DIMENSION QF(5,16,3),QC(4,13,3),DS(4,13) DIMENSION VORTIC(50,13),QW(50,14,3),A1(4,13),ALAMDA(2) DIMENSION VORT1(50,13),QW1(50,14,3) COMMON VORTIC,QW,VORT1,QW1,QF,A1 COMMON IT,ALF,SNO,CSO,BB,QC,DS,ALAMDA,DXW C C C C C PAY=3.141592654 IB:NO OF CHORDWISE BOXES, IB1=IB+1 JB1=JB+1 DO I=1,IB1 SN(I)=SIN(ALF(I)) CS(I)=COS(ALF(I)) CTG1=TAN(PAY/2.-ALAMDA(1)) CTG2=TAN(PAY/2.-ALAMDA(2)) CTIP=C+B*(CTG2-CTG1) S=B*(C+CTIP)/2 AR=2.*B*B/S WING FIXED VORTICES LOCATION JB:NO OF SPANWISE BOXES BJ=0 DO J=1,JB1 IF(J.GT.1) BJ=BJ+BB(J-1) Z1=0 DC1=BJ*CTG1 DC2=BJ*CTG2 DX1=(C+DC2-DC1)/IB ( QF(I,J,(X,Y,Z)) ) D.3 C C C C C C C C C Time-Dependent Programs DC1=LEADING EDGE X, DC2=TRAILING EDGE X DO I=1,IB QF(I,J,1)=DC1+DX1*(I-0.75) QF(I,J,2)=BJ QF(I,J,3)=Z1-0.25*DX1*SN(I) Z1=Z1-DX1*SN(I) THE FOLLOWING LINES ARE DUE TO WAKE DISTANCE FROM TRAILING EDGE QF(IB1,J,1)=C+DC2+DXW QF(IB1,J,2)=QF(IB,J,2) QF(IB1,J,3)=Z1-DXW*SN(IB) WING COLLOCATION POINTS DO J=1,JB Z1=0 BJ=QF(1,J,2)+BB(J)/2 DC1=BJ*CTG1 DC2=BJ*CTG2 DX1=(C+DC2-DC1)/IB DO I=1,IB QC(I,J,1)=DC1+DX1*(I-0.25) QC(I,J,2)=BJ QC(I,J,3)=Z1-0.75*DX1*SN(I) Z1=Z1-DX1*SN(I) DS(I,J)=DX1*BB(J) ROTATION OF WING POINTS DUE TO ALFA SN1=SIN(-ALFA) CS1=COS(-ALFA) DO I=1,IB1 DO J=1,JB1 QF1=QF(I,J,1) QF(I,J,1)=QF1*CS1-QF(I,J,3)*SN1 QF(I,J,3)=QF1*SN1+QF(I,J,3)*CS1 IF((I.EQ.IB1).OR.(J.GE.JB1)) GOTO QC1=QC(I,J,1) QC(I,J,1)=QC1*CS1-QC(I,J,3)*SN1 QC(I,J,3)=QC1*SN1+QC(I,J,3)*CS1 CONTINUE RETURN END C C C C C THE FOLLOWING SUBROUTINES ARE LISTED WITH THE STEADY STATE VORTEX LATTICE SOLVER (PROGRAM No 13) SUBROUTINE VORTEX(X,Y,Z,X1,Y1,Z1,X2,Y2,Z2,GAMA,U,V,W) C SUBROUTINE DECOMP(N,NDIM,A,IP) C C SUBROUTINE SOLVER(N,NDIM,A,B,IP) 609 Index Acceleration of fluid particle, Added mass, 192–194, 385–387, 398 Aerodynamic center, 110 Aerodynamic loads, 85–87 Aerodynamic twist, 181 Airfoil circular arc, 134–135, 138–139 Joukowski, 135–137, 139–140 multielement, 311–312 NACA nomenclature, 499 van de Vooren, 137–138, 140–141 Angle of attack definition, 75 effective, 171–172 induced, 171–172 zero lift, 109 Angular velocity, 21–22 Aspect ratio, 175 Barrier, 29 Bernoulli’s equation, 28–29 Biot-Savart Law, 36–41 Blasius formula, 128 Body forces, Bound vortex, 89–90 Boundary conditions Dirichlet, 49, 208–209 inviscid flow, 18, 27–28 Neumann, 49, 207–208 no slip, 11 small-disturbance, 76–78 solid surface, 11 unsteady flow, 372–373 Boundary layer (Laminar) Blasius solution, 461–463 classical equations, 18–19, 448–452 displacement thickness, 454–455 far wake solution, 472–473 friction coefficient, 465 Goldstein singularity, 471–472 integral kinetic energy equation, 474 integral kinetic energy shape factor equation, 474–475 Karman-Pohlhausen method, 468–469 momentum thickness, 465 second-order equations, 452–456 shape factor, 465 similar solutions, 457–459 611 stagnation point solution, 461–462 Thwaites method, 469–471 viscous-inviscid interaction, 475–480 von Karman integral momentum equation, 463–467 Camber function, 78 Canard, 485–486 Cauchy integral theorum, 123 Cauchy principal value, 98 Cauchy-Riemann conditions, 42, 123 Center of pressure, 109 Circular cylinder flow lifting, 65–66 non-lifting, 62–65 Circulation definition, 23 rate of change, 25–26 Collocation point, 115 Complex plane, 142 Complex variable approach circle plane, 128–130 complex potential, 125–126 complex velocity, 126 conformal mapping, 125, 128 Joukowski transformation, 128–137 van de Vooren airfoil, 137–138 Composite expansion, 161–162 Compressibility, 19, 90–92, 226–227 Coning motion, 443–445, 527–528 Continuity equation, 7–9, 11–12 Coordinate systems cartesian, cylindrical, 11 spherical, 12 Corner, flow in, 55–56, 127 Cosine panel spacing, 277–278 Crossflow plane, 185 Crow instability, 486–487 Cusped trailing edge, 137, 211–212, 325–327 d’Alembert’s paradox, 107–108 Del (gradient) operator cartesian coordinates, cylindrical coordinates, 11 definition, spherical coordinates, 12 Delta wing, 192, 404–407, 516–523 Dihedral, 350–351 612 Dimensional analysis, 17–19 Dirichlet boundary condition, 49, 208–209 Divergence theorem, Divergence, vector field cartesian coordinates, cylindrical coordinates, 12 spherical coordinates, 12 Doublet distribution, 47–48, 72, 235–236, 239–241, 242–244 quadrilateral, 247–250 three-dimensional, 47–48, 51–54 two-dimensional, 48, 57–58, 231 Downwash definition, 170–171 lifting-line, 169–172 Drag definition, 86 friction drag, 506 induced drag, 173–175, 201–204 Drag coefficient, 69, 87 Effective angle of attack, 171–172 Ellipse, flow past, 99–100 Elliptic lift distribution, 173–178 Euler angles, 369 Euler equation, 11 Euler number, 16 Eulerian method, Flap, 113–114, 298, 501–505 Flat plate airfoil, 110–112, 130–131 in normal flow, 133–134 oscillation of, 396–399 sudden acceleration of, 381–387 Flow similarity, 19 Fluid element, 1, Force, 4–6 Fourier series, 106 Free stream, 54 Free-surface flows, 530–533 Froude number, 15–16 Fuselage, effect on lift, 360–361 Gap, effect on wing, 364–366 Geometric twist, 181 Glauert integral, 98 Green’s theorem, identity, 30, 44–45 Ground effect inclusion in computation scheme, 338–340 lumped vortex, 116–118 rectangular wings, 350 unsteady flow, 431, 433 Helmholtz vortex theorems, 34 Horseshoe vortex, 168–171, 256–258 Index Images, method of circle, 144–146 parallel walls, 142–144 plane wall, 141–142 Incompressible fluid, Induced angle of attack, 171–172 Induced drag, 173–175, 201–204, 336–338, 346–347 Infinity condition, 28 Influence coefficient definition, 214 Irrotational flow, 23 Jones, R T., method of, 192–194 Joukowski transformation, 128–137 Kelvin theorem, 25–26 Kinematic viscosity, 16 Kutta condition, 88–89, 209–213, 375–376, 416–419 Kutta-Joukowski theorem, 66–67, 128 generalized, 146–149 Lagrangian method, Laminar bubble, 496, 499 Laplace’s equation cylindrical coordinates, 11 definition, 27 spherical coordinates, 12 Leading edge separation, 496, 516–528 Leading edge suction, 107–108, 131–133 Lift, 86 Lift coefficient definition, 69, 87 for finite wings, 175 maximum, 497, 504 for thin airfoils, 109 Lift slope, 109–110, 175–176 Lifting line, 167–183, 331–338 Lifting surface, 82–85 numerical, 340–351 unsteady, 479–491 Local (leading edge) solution, 157–160 Lumped vortex element, 134–135 Mach number, 16 Mapping, 125 Matched asymptotic expansions, 160–163 Material derivative, 9, 11–12 Milne-Thomson circle theorem, 144–146 Moment coefficient, 87 Momentum equation, 7–13 Multielement wing, 363–364, 501–504 Navier-Stokes equations, 10–13 Neumann boundary condition, 30, 49, 207–208 Newton’s second law, 7, 10 Normal force, 86 Normal stress, No-slip condition, 11 Index Panel methods, 206–226, 262–367 Parabolic arc airfoil, 112–113, 268–270 Pathlines, Perturbation methods, 151–166 Poisson’s equation, 36 Prandtl-Glauert rule, 91–92 Pressure, Pressure coefficient, 16, 430 Propulsion effects, 528–530 Rankine’s oval, 60–62 Reduced frequency, 373, 418–419 Residue theorem, 124 Reynold’s number, 16 Rotor, 379, 504–506 Separated flow, 69, 508–516 Separation point, 509, 511 Shear stress, Side force, 86 Similarity of flow, 19 Slat, 534–535 Slender body theory, 195–201 Slender wing theory, 184–195 in unsteady motion, 400–407 Source distribution, 47–48, 70–72, 233–234, 238–239 quadrilateral, 245–247 three-dimensional, 47–51 two-dimensional, 48, 56–57, 230–231 Sphere, flow past, 67–69 Stagnation flow, 55–56 Stagnation point, 56 Stall, 536–537, 542 Starting vortex, 168, 364–365 Stokes theorem, 22 Streak lines, Stream function, 41–43 Streamline, 3–4 Stress vector, 4–6 Strouhal number, 15, 539 Sudden acceleration flat plate, 381–387 rectangular wing, 429–431 Superposition principle, 60 Surface forces, 613 Swept wings, 347–349 Symmetric airfoil, 94–100 Symmetric wing, 79–82 Tandem airfoils, 116 Tangential stress, Taper ratio, 349–350 Theodorsen lift deficiency function, 398–399 Thickness function, 78 Thin-airfoil theory, 94–121 Transition, 523, 526 Transpiration velocity, 227–228, 476, 491–492 Trefftz plane, 202–204 Turbulent boundary layer, 487–495 Twist, 172, 181–183 Uniqueness of solution of Laplace’s equation, 30–32 van de Vooren airfoil, 137–138, 140–141 Velocity, 1–2 Velocity potential, 26 perturbation, 77 total, 77 Viscosity coefficient, Vortex Asymmetry, 520–522 Burst (breakup), 520–522 core, 36, 254–255 distribution, 73, 236–237, 241–242 filament, 32–34 horseshoe, 168–171, 296–297 irrotational, 36, 58–60 lift, 517–522 line, 32, 38–41, 251–255 ring, 250–251, 255–256 two-dimensional, 34–36, 231–232 Vorticity definition, 22 rate of change, 24–25 Wake, 83–85, 87–90, 364, 508–509 Wake rollup, 483–487, 512–514 Wind tunnel wall interference, 118–119, 163–165, 363–364, 530 Zero-lift angle of attack, 109 ... = σ0 (x − x1 ) ln r 12 − (x − x2 ) ln r 22 + 2z(? ?2 − θ1 ) 4π σ1 x − x 12 − z x − x 22 − z + ln r 12 − ln r 22 4π 2 + 2x z(? ?2 − θ1 ) − x(x2 − x1 ) (panel coordinates) (11.108) Of particular interest... ⎢−0. 02 0.50 0.01 0.01 0.01 0.01 ⎢ ⎢−0. 02 0.01 0.50 0. 02 0. 02 0.03 ⎢ ⎢−0.01 0.01 0. 02 0.50 0.04 0.08 ⎢ ⎢ ⎢ 0.00 0.01 0. 02 0.06 0.50 0 .24 ⎢ ⎢ 0.01 0.01 0.03 0. 12 0 .24 0.50 ⎢ ⎢ 0. 02 0.03 0.10 0 .23 ... (11. 62) , specified for each collocation point → N , the matrix equation will have the form ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ a11 , a 12 , , a1N μ1 b11 , b 12 , , b1N σ1 ⎜ a21 , a 22 , , a2N ⎟⎜ ? ?2 ⎟ ⎜ b21 , b22