Energy Decay of Vortices in Viscous Fluids: an Applied Mathematics View

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1 Energ Deca of Vortices in Viscous Fluids: an Applied Mathematics View Jan Nordström and Björn Lönn Linköping Universit Post Print N.B.: When citing this work, cite the original article. Original Publication: Jan Nordström and Björn Lönn, Energ Deca of Vortices in Viscous Fluids: an Applied Mathematics View, 1, Journal of Fluid Mechanics. Copright: Cambridge Universit Press (CUP) Postprint available at: Linköping Universit Electronic Press

2 Under consideration for publication in J. Fluid Mech. 1 Energ Deca of Vortices in Viscous Fluids: an Applied Mathematics View J A N N O R D S T R Ö M 1 A N D B J Ö R N L Ö N N 1 Department of Mathematics, Linköping Universit, SE81 83 Linköping, Sweden Department of Information Technolog, Uppsala Universit, SE-71 Uppsala, Sweden (Received?; revised?; accepted?. - To be entered b editorial office) The energ deca of vortices in viscous fluids governed b the compressible Navier- Stokes equations is investigated. It is shown that the main reason for the slow deca is that zero eigenvalues eist in the matri related to the dissipative terms. The theoretical analsis is purel mathematical and based on the energ method. To check the validit of the theoretical result in practice, numerical solutions to the Navier-Stokes equations are computed using a stable high order finite difference method. The numerical computations corroborate the theoretical conclusion. Ke words: 1. Introduction Vortices are common flow features in aeronautical applications. The topic of this paper, the slow energ deca, causes problems when wingtip vortices remain on the runwa for a long time, see Proctor (1998), Kantha (1), Gerz et al. (), Breitsamter (11), Spalart (1998). Vortices are also important when eplaining several flow phenomena. The lift generated b wings is investigated in Chen et al. (1), Rossow (1999), Yen (11). It is concluded that effects from leading edge vortices can be used to increase lift. Vortices are frequentl present in nature and also influence the environment. The presence of von Karman vortices in the trailing wind of islands can have effects on the local climate as well as the sea life, see Beggs et al. (), Ponta (1), Holzapfel et al. (3). Their effect on fish, bird and insect behavior is studied in Wu (11), Kim & Kim (11), Liu et al. (9). Vortices are also needed when modeling the general circulation of the ocean as seen in Holland (1978), Zhang et al. (11), Winter & Bourqui (11), Koszalka et al. (11). Other effects include emissions from air traffic Gerz & Holzapfel (1999) and influence on the efficienc of turbine parks Magnusson & Smedman (1999). It is well known that vortices do not deca in inviscid flows governed b the Euler equations. It is quite straightforward to construct eact solutions that advect with a constant background velocit, see Anderson (1991), McCormack & Crane (1973), Erlebacher et al. (1997), Davoudzadeh et al. (199). Less well known is wh this slow energ deca seem to persist also in viscous flows governed b the Navier-Stokes equations. This phenomena has been thoroughl investigated and man theories of what phenomena are important and wh the deca is slow eist, see for eample Sarpkaa (1998), Matthaeus et al. (1991), Gustavsson (1991), Greene (198), Balmforth et al. (1), Greene (198), Zhou et al. (1999), Wallin & Girimaji (), Pullin & Saffman (1998), Winckelmans et al. (), address for correspondence: jan.nordstrom@liu.se

3 J. Nordström and B. Lönn Kundu & Cohen (1). Related but more fundamental theoretical questions like stabilit and eistence of solutions for long times were studied in Hoff & Zumbrum (199), Hagstrom & Lorenz (199) and Hagstrom & Lorenz (). Most of the previous investigations and theories are based on observations of measured or computed data and a subsequent analsis. We will take a different approach and rel on mathematics onl. This will eliminate potential error sources when producing data, make the analsis transparent to the reader and hence enable the reader to follow and check the whole procedure. Onl afterwards, when the main theoretical conclusion has been reached, and the suggested deca scenario has been formulated will the implications of the theor be tested using results from numerical calculations. We begin b observing that vortices eist for all ranges of Renolds numbers (with the eception of Stokes flow) including inviscid flows. This indicate that the size of the viscous terms is not of fundamental importance. Hence, it is relevant to analse a simple laminar flow case, without considering complicated additional phenomena such as turbulence. We will analse the energ deca of small scale disturbances b appling the standard tool in applied mathematics/numerical analsis, namel the energ-method (see Kreiss & Lorenz (1989) and Gustafsson et al. (199) and the numerous references therein). In particular we will investigate the viscous terms and their role in the energ dissipation.. Analsis Consider the flow situation when small disturbances are imposed on a constant background state. This could be a model of the situation right after the passing of an aircraft in free space. This problem setup also naturall allows for the linearized analsis below where we neglect quadratic terms and investigate the deca of small scale disturbances. The dependent variables are the densit ρ, the velocities in each direction ũ, ṽ, w and the temperature T. The tilde sign signifies the presence of dimensions. We write the equations in non-dimensional form using the free stream densit ρ, the free stream velocit Ũ and the free stream temperature T. The shear and second viscosit coefficients µ, λ as well as the coefficient of heat conduction κ are non-dimensionalized with the free stream viscosit µ. The pressure in non-dimensional form becomes: p = p/( ρ Ũ ) = ρt/(γm ). Also used are M = Ũ γ R T, P r = µ C p κ, Re = ρ Ũ L µ, γ = C p C v, ϕ = γκ P r (.1) where L is a length scale and M, P r, Re = 1/ɛ and γ are the Mach-, Prandtl-, Renolds number and ratio of specific heats respectivel. The time scale is L/Ũ..1. The energ method The linearized dimensionless compressible Navier-Stokes equations can formall be written t + Ā + B + C z = ɛ ρ [( D 11 + D 1 + D 13 z ) + +( D 1 + D + D 3 z ) + ( D 31 + D 3 + D 33 z ) z]. (.) In (.), V = (ρ, u, v, w, T ) T is the disturbance and the constant state around which we linearize is denoted b an overbar. The matrices related to the hperbolic terms in (.) must be smmetric for the energ method to be applicable Nordström & Svärd ().

4 Energ Deca of Vortices in Viscous Fluids 3 In Abarbanel & Gottlieb (1981) it is shown that the smmetrization can be done using a single matri S. We choose the smmetrizer [ ] S 1 c ρ = diag, ρ c, ρ c, ρ c,, (.3) γ γ(γ 1)M where c is the speed of sound at reference state. Remark: Abarbanel & Gottlieb (1981) showed that the three-dimensional compressible Navier-Stokes equations with 1 matrices involved (see (.)), can be smmetrized b a single matri (smmetrizer). This remarkable fact was complemented b the observation that there are at least two different smmetrizers, based on either the hperbolic or the parabolic terms. The smmetrizer (.3) is related to the parabolic terms. The similarit transform is applied b multipling (.) from the left with S 1. After appling the smmetrizer we obtain ū c/ γ c/ γ ū c γ Ã = ū ū c γ 1 γ ū γ 1 D 11 = µ + λ µ µ ϕ (.) B = v c/ γ v c/ γ v c γ 1 γ v c γ 1 γ v D = µ µ + λ µ ϕ (.) C = w c/ γ w w c/ γ w c c γ 1 γ γ 1 γ w D 33 = µ µ µ + λ ϕ (.) D 1 = D 1 T = λ µ D 13 = D 31 T = λ µ (.7) D 3 = D 3 T = λ µ Ṽ = c γ ρ ρ cu ρ cv ρ cw ρ γ(γ 1)M T (.8) where Ṽ T = (S 1 V ) T, Ã = S 1 ĀS, B = S 1 BS, C = S 1 CS and the Dij s are

5 J. Nordström and B. Lönn unchanged. The matrices above are all given in Abarbanel & Gottlieb (1981) but we list them here for completeness. The energ method is applied b multipling the smmetrized dimensionless Navier- Stokes equations with Ṽ T and integrating over the domain Ω. Let the energ norm be defined as Ṽ = ΩṼ T Ṽ dddz. B using Gauss theorem and integration b parts, we obtain Ṽ t + BT = ( ɛ ρ ) Ω Ṽ Ṽ Ṽ z T D 11 D1 D13 D 1 D D3 D 31 D3 D33 Ṽ Ṽ Ṽ z dddz, (.9) where BT = Ω Ṽ T [ÃṼ ( ɛ ρ )( D 11 Ṽ + D 1 Ṽ + D 13 Ṽ z )] Ṽ T [ BṼ ( ɛ ρ )( D 1 Ṽ + D Ṽ + D 3 Ṽ z )] Ṽ T [ CṼ ( ɛ ρ )( D 31 Ṽ + D 3 Ṽ + D 33 Ṽ z )] ˆn ds. (.1) In (.1), ds = d + d + dz and ˆn = (n 1, n, n 3 ) T is the outward pointing unit normal on the surface Ω. In this paper we ignore the effect of far-field boundaries and consider the Cauch problem onl. In particular, we focus on the volume term on the right-hand-side of (.9) and neglect the boundar term BT. Let the block matri consisting of the D ij matrices be denoted b D. If D is positive semi-definite, (.9) show that the energ decas. Remark: Normall the relations (.9) and (.1) are used in order to determine boundar conditions that lead to well-posedness, see for eample Gustafsson & Sundström (1978), Nordström (199) and Nordström & Svärd (). In that case the focus is on the term BT. Remark: If D would not be positive semi-definite, then (.9) would show that the Cauch problem for the linearized Navier-Stokes equation is not well posed. The first, sith and 11th rows as well as the first, sith and 11th columns of the 11 matri D have onl zero elements, thus the matri can be reduced to a 11 matri. We refer to this matri as ˆD. µ + λ λ λ µ µ µ µ ϕ µ µ ˆD = λ µ + λ λ µ µ (.11) ϕ µ µ µ µ λ λ µ + λ ϕ The eigenvalues of ˆD are obtained b solving det( ˆD si) = for s which ield s = ϕ, ϕ, ϕ, µ, µ, µ,,,, µ, µ, µ + 3λ. The second law of thermodnamics, see White (197), impl that µ + 3λ and hence all the eigenvalues are positive semi-definite as required for a well posed problem. To see how the velocit gradients and the eigenvalues are related, the eigenvectors

6 Energ Deca of Vortices in Viscous Fluids must be calculated. The orthonormal matri with the eigenvectors as columns is Q = 3. (.1) We obtain ˆD = QΛQ T where Λ = diag(ϕ, ϕ, ϕ, µ, µ, µ,,,, µ, µ, µ + 3λ). B inserting ˆD (or the identical D) into (.9) we obtain the final form of the energ deca rate T Ṽ t = ( ɛ ρ Ṽ Ṽ ) Q T Λ Q T dddz, (.13) where Q T Ṽ D Ṽ Ṽ z = Ṽ Ṽ z ρ T γ(γ 1)M ρ T γ(γ 1)M ρ T γ(γ 1)M z Ṽ Ṽ z 1 ρ c[v + u ] 1 ρ c[w + u z ] 1 ρ c[w + v z ] 1 ρ c[v u ] 1 ρ c[w u z ] 1 ρ c[w v z ] 3 ρ c[ 1 u v + 1 w z] 1 ρ c[u w z ] 1 3 ρ c[u + v + w z ]. (.1).. Analsis of the energ deca rate The relation (.13) and (.1) show the relation between eigenvalues and gradients. The temperature gradients are multiplied b the eigenvalue ϕ. The following three components consist of terms related to the rotation of the velocit field in the -, z- and z planes and are each multiplied b µ. The most interesting fact is that the vorticit components in each plane are multiplied b a zero eigenvalue. The last term is the divergence and it is multiplied b the positive eigenvalue µ + 3λ. The two remaining terms consist of components related to the divergence u + v + w z and the are multiplied b µ. Based on (.13) and (.1) we can now draw the two central conclusions of this paper:

7 J. Nordström and B. Lönn The vorticit terms are multiplied b zero eigenvalues. The deca rate is proportional to the gradients squared. These conclusions suggest that vortices will deca slowl because the main part of the energ is concentrated in the vorticit components. The also indicate that flow structures with gradients that are not in vorticit form will deca fast. Remark: The velocit gradient can be decomposed into smmetric and antismmetric parts (strain-rate and vorticit tensors). The eigenvalues associated with the vorticit tensor are zero while eigenvalues associated with the strain-rate tensor are non-zero. The onl flow condition that has zero strain rate and non-zero vorticit is solid bod rotation, which, according to the analsis, is the onl flow that will be undamped. All other kind of vortical flows will be damped since it will contain some amount of strain rate. Remark: The analsis above show that if all the energ reside in the vorticit components, there is no energ deca. However, the energ can be transferred to other modes and lead to deca. If the transfer is slow, the energ deca will be small. On the other hand it is also possible to have an energ deca that halts due to the transfer of energ from non-vorticit modes into vorticit modes. It is not possible (although indications eist, see Hoff & Zumbrum (199)) to analticall determine whether such a transfer eist and how fast it is, but we will investigate this below b performing numerical calculations. 3. Numerical calculations The computations are done using a well tested and provable stable high order finite difference scheme for the compressible Navier-Stokes equations. It uses summation-bparts operators and imposes boundar and interface conditions weakl as described in Carpenter et al. (1999), Nordström & Carpenter (1999), Nordström & Carpenter (1), Mattsson et al. (), Svärd et al. (7), Svärd & Nordström (8) and Nordström et al. (9). The code runs with,3, and th order global accurac. The 3rd order accurate scheme with far field boundaries sufficientl far from the actual flow structure was used The grids and order of accurac The grid used for calculations in two dimensions is each divided into 1 blocks. All blocks have an equal number of grid points. The grid used, for the displaed computations, is a square with side length and 7 points each in both the and directions. Hence the mesh has = =.78. For calculations in three dimensions the grid is a cube with side length and it has 19 points in each direction giving = = z =.1. The 3rd order accurate scheme was evaluated in two dimensions using the , and point grids. The practical order of accurac obtained was 1.71,.18 and. respectivel. The error level in L norm for the displaed D computations is estimated to Initial conditions in two dimensions To verif the analtical results, long time simulations with three different initial conditions have been investigated. The Renolds numbers considered are Re = 1, 1 and Re = 1. We stress here that the size of the Renolds number is of no fundamental importance in this investigation. The zero eigenvalues are present for all Renolds numbers. We have compared the energ deca of a vorte, with the energ deca of white noise, and a speciall constructed divergence free flow case. As initial values for all these two dimensional calculations we used: the Mach number M=., the radius r=1, the densit and temperature are ρ = 1, T = 1.

8 Energ Deca of Vortices in Viscous Fluids Figure 1: The initial field for the vorte. The velocities decrease fast outside r=1. The vorte is a so called wake vorte also used in Winckelmans et al. (). The initial velocit field is shown in Figure 1. It is defined b ε ij ε ij u ij = ij + ij + v ij = r c ij + ij + (3.1) r c where ij, ij are the distances from the center of the vorte in each direction, r c is the distance from the center to the strongest velocit of the vorte and ε = r c M. For completeness we also give the velocit components in clindrical coordinates as U θ = εr r + rc U r = (3.) where r = + is the radius and U θ, U r are the tangential and radial velocities respectivel. The initial velocit field of the white noise is shown in Figure. It is defined b u ij = ε random ij + ij + r c v ij = ε random ij + ij + r c rand [ 1, 1] (3.3) where ɛ is the same as in (3.1). The velocities are randomized using random numbers from the Fortran 9 random subroutine and damped to zero using the same function as the vorte condition. The velocities in clindrical coordinates are U θ (r, θ) = ε random r + rc, U r (r, θ) = ε random r + rc (3.) with the same definitions as in (3.). In all the calculations in this paper, the Mach number is small, and the initial condition should (almost) satisf the zero divergence relation in incompressible flow. However, the

9 8 J. Nordström and B. Lönn Figure : The initial field for the white noise. The velocities decrease fast outside r= Figure 3: The initial field for the divergence free case. The velocities decrease fast outside r=1.

10 Energ Deca of Vortices in Viscous Fluids (a) t= (b) t=1 1 (c) t= Figure : Time evolution of the vorte for Re=1. The radius is displaed b the low densit core indicated with dark blue. As time passes, the velocities slowl weaken. white noise do not have zero divergence. In order to check if that has an impact on the results, we included a special zero divergence initial condition shown in Figure 3 of the following form uij = ε ij + ij ep [(ij + ij ) + rc ] vij = ε ij + ij. ep [(ij + ij ) + rc ] (3.) Here ε is the same as in (3.1). When r > 1 we replace the denominator ep [(ij + ij ) + + rc ] for a fast deca of the solution. The formulation (3.) in rc ] with ep [ij + ij clindrical coordinates is Uθ (r, θ) = εh(r, θ)cos(θ + 3π/), Ur (r, θ) = εh(r, θ)sin(θ + 3π/) (3.) where h(r, θ) = r(cos(θ) + sin(θ))/ ep(r (cos(θ) + sin(θ)) + rc ) Time evolution The time evolution of the vorte can be seen in Figure. The kinetic energ of the vorte spread evenl to the area surrounding it. At t= the vorte seems onl slightl weaker. In an isolated vorte one has an inner vorte core (r < rc ) which is rotating as a solid bod. In the outer part (r > rc ), both the strain rate (with non-zero eigenvalues) and the vorticit is present, see the first remark in section.. Thus, the vorte will be damped from outside leaving the vorte core undamped but with a shrinking core radius. This is eactl what is seen in Figure, and also consistent with the finding of Wallin & Girimaji () for turbulent deca of wing tip vortices. The white noise computation is displaed in Figure. It decas quickl and has almost vanished at t=1. A closer look at what remain of the white noise at t= and t=1 reveal several vorticit like flow structures. The time development of the special divergence free case is displaed in Figure. The flow develops into two counter clockwise rotating vortices. These vortices remain for a long time and deca slowl. Remark: The result of these three calculations is consistent with the theor in Section. The fact that even if one starts the calculation with a non-vorte structure (as in Figures and ), one ends up with onl vorte like structures is particularl striking. This could be a scenario in which energ is transferred from non-vorticit modes to vorticit modes as mentioned earlier.

11 J. Nordstro m and B. Lo nn (c) t= (b) t= (a) t=8 - (d) t= zoomed - (e) t=1 zoomed Figure : Time evolution of the white noise for Re=1. The white noise decas quickl. At t= and t=1 onl vorticit like flow structures remain. 3.. Energ deca The kinetic energ of the global sstem E = ρ u +v +w as a function of time, normalized b the initial kinetic energ, is shown in Figure 7. The energ deca of white noise is clearl faster than the energ deca of the wake vorte and the divergence free flow case. The fast deca is predicted b the theor since white noise has large gradients initiall compared to the other initial conditions. Note that once the white noise has developed into a flow consisting mainl of vorticit, see Figure, the deca stops. The energ of the vorte decrease slowl with time which agrees with the results obtained in Winckelmans et al. (). Despite the fact that the wake vorte flow with Re=1 looks weaker at t= more than 9% of the kinetic energ in the global sstem remain. The energ deca of the divergence free case is faster than the vorte energ deca but slower than the white noise energ deca. The deca is semi-fast initiall when the velocit

12 Energ Deca of Vortices in Viscous Fluids 1 (a) t= (b) t=1 1 (c) t= Figure : Time evolution of the divergence free case for Re=1. A wave leaves the domain at t=1. The remaining counter clockwise rotating vortices deca slowl. 1.9 Normalized Kinetic Energ (E/E).8 Wake vorte Re=1 Wake vorte Re=1 Wake vorte Re=1 Div. Free Re=1 Div. Free Re=1 Div. Free Re=1 White noise Re=1 White noise Re=1 White noise Re= Dimensionless Time (t) Figure 7: Normalized kinetic energ deca for the D vorte, divergence free and white noise cases as a function of time. The vorte has the least energ deca. gradients are strong and decreases as the flow becomes dominated b the two counter rotating vortices. Once the vortices have formed, the deca slow down significantl. For all three cases, computing with a lower/higher Renolds number consistentl resulted in more/less energ deca for all three cases. Remark: The result of the calculations, i.e that vorte like structures deca slowl and other flow structures relativel fast is consistent with the theor in Section. The results are also consistent with the asmptotic deca rates obtained in Hoff & Zumbrum (199) for the Cauch problem, if one takes into account the fact that we are computing on a finite domain. (On a finite domain, the acoustic part will radiate to infinit and deca rapidl). The results of Hoff & Zumbrum (199) also indicate that the transfer of energ between the vorticit modes and other modes is small, which again is consistent with our computational results.

13 1 J. Nordström and B. Lönn Normalized Kinetic Energ (E/E ) Wake vorte Re=1 Div. Free Re=1 White noise Re= Dimensionless Time (t) Figure 8: Normalized kinetic energ deca for the 3D vorte, divergence free and white noise cases as a function of time. The vorte has the least energ deca also in 3D. 3.. Validation of the results To confirm the results obtained in two dimensions, also three dimensional calculations were made. The initial velocities are defined as: V orte : u ijk = ε ijk f(,, z), v ijk = ε ijk f(,, z), w ijk = (3.7) W hite noise : u ijk = ε random f(,, z), v ijk = ε random f(,, z), w ijk = (3.8) Divergence free : u ij = ε ij + ij g(,, z), v ij = ε ij + ij g(,, z), w ijk = (3.9) where f(,, z) = ijk + ijk + z ijk + r c and g(,, z) = ep [( ij + ij + z ij ) + r c]. The same parameter values as in two dimensions were used. Note that the flow is trul three dimensional due to the z-dependence in f(,, z) and g(,, z) even though the velocit in the z-direction is initiall zero. Note also that (3.9) is, strictl speaking, no longer divergence free due to the z dependence. The focus of the computations in three dimensions is to verif the relevance of the computations in two dimensions. Far field boundar conditions are used on all outer boundaries, ecept in the z-direction where periodic boundar conditions are used. The energ of the global sstem as a function of time is shown in Figure 8. In all three computations the energ deca is similar to and hence validate the energ deca obtained from the two dimensional computations.. Discussion The main result of the mathematical analsis, relation (.13), describes how the disturbance energ (the deviation from the constant background field) deca (or do not deca). The right hand side of (.13) is diagonalised such that each eigenvalue multiplies a quadratic form. Each quadratic form, shown in (.1), consists of linear combinations

14 Energ Deca of Vortices in Viscous Fluids 13 of the gradients of the flow variables. The eigenvalues and quadratic forms can be organized in two groups. Group one has non-zero positive eigenvalues and group two has zero eigenvalues and correspond eactl to the vorticit modes. For eas of eplanation we denote the modes in group one, the deca modes. We start b considering the energ deca onl. Firstl, assume that all the quadratic forms/modes are non-zero. Due to the non-zero eigenvalues related to the deca modes, we have energ deca. Secondl, assume that the deca modes are zero, while the vorticit modes are non-zero. Then, since the eigenvalues related to the vorticit modes are zero, the energ will sta constant even though the corresponding quadratic forms/modes are non-zero. Note that if all the eigenvalues were non-zero, no possibilit for a constant energ would eist. Now we include energ transfer in the discussion. Consider the second case above which at a particular time do not ield energ deca, i.e. all the energ reside in the vorticit modes. If the energ from the the vorticit modes is transferred to the deca modes, we have energ deca in the net instant. If the transfer is large, we will have a large deca, if the transfer is small we will have a small deca. The reversed scenario is also possible, i.e. that energ ma transfer from deca modes to vorticit modes which will decrease the energ deca from a high level to a low level. The results discussed above have implications for the practical treatment (both the removal and the preservation) of energ content in flow structures. For a general flow structure from which one wants to remove energ, one could aim for transferring energ from the vorticit modes into the deca modes, and use the natural deca mechanism discovered in this paper as guidance.. Conclusions We have analsed the energ deca of small scale disturbances in laminar flow b using the energ-method. An equation for the energ deca rate in terms of gradients and a dissipation matri was derived. Zero eigenvalues of the dissipation matri multiplied the vorticit components. The zero eigenvalues multipling the vorticit components suggest that flows with most of the energ in the vorticit components deca slowl. It also mean that small scale disturbances will have most of their energ in vorticit form after a long time. Numerical simulations verified that the energ deca is slow for vortices compared to the white noise and divergence free flow cases. The simulations also verified that the energ transfer from the vorticit components to other modes is slow, leading to a slow overall deca. The eistence of zero eigenvalues multipling the vorticit component of the flow is probabl one of the main reasons for the slow deca of energ in vorte like structures in fluid mechanics. The energ transfer between vorticit modes and non-vorticit modes was investigated numericall. This part could possibl be improved b an even more advanced analsis and we hope that the finding in this paper will inspire others to do that. REFERENCES Abarbanel, S. & Gottlieb, D Optimal time splitting for two- and three-dimensional Navier-Stokes equations with mied derivatives. Journal of Computational Phsics 1 (1), Anderson, J. D Fundamentals of Aerodnamics, nd edition. McGraw-Hill, Inc.

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