An Informal Introduction Into the Basic Concepts of Aerodynamics Part I

Int. J. Aerodynamics, Vol. X. No. Y, xx.u

An informal introduction to basic concepts of aerodynamics: part I: incompressible two-dimensional flows

M. Hafez Department of Mechanical and Aerospace Engineering. University of California, Bainer Hall. One Shields Ave .. Davis, CA 95616-5294, USA E-mail: [email protected]

Abstract: The main concepts to understand and to evaluate aerodynamic forces and moments are examined using as simple mathematical tools as possible. The lifting and thickness problems for 2-D incompressible flows are discussed. In particular, Magnus effect in flow over a rotating cylinder is analysed together with Joukowski transformation and Joukowski airfoils. Viscous effects are also briefly examined.

KeywordS: flow over rotating cylinders; Joukowski transformation and airfoils; boundary layers.

Reference to this paper should be made as follows: Hafez, M. {xxxx) 'An infonnal introduction to basic concepts of aerodynamics: part I: incompressible two-dimensional flows'. Int . .J. Aerodynamics. Vol. X, No. Y, pp.000-000. Biographical notes: M Hafez received his PhD from University of Southern California. Department of Aerospace Engineering in 1972. Then he worked at Flow Research Inc. in Kent Washington and at NASA Langley Research Center in Hampton Virginia, before he joined University of California. Davis in 1985 as a ~ofessor of Aeronautical Engineering. His fields of interest are transonic aerodynamics and computational fluid dynamics (CFD).

1 Introduction

There are many text books and monographs available to study aerodynamics in a rigorous manner (Glauert. 1926; Munk, 1929; Prandtl and Tietjens, 1934a, 1934b; Pope, 1951; Milne-Thomson, 1958, 1968; Kochin et al., 1964; Ashly and Landhal, 1965; Karamcheti, I 966; Duncan et al . 1972; Krasnov, 1978; Anderson. I 979; Bertin and Cummings, I 979; Moran, 1985; Lighthill, 1986; Shevell. 1989; Jones, I 990~ Clancy, 1996; Smetana, I 997; Keuthe and Chow, 1998; Marshall, 200l;Katz and Plotkin, 2001; Dragos, 2003; White, 2008; Anderson, -2007; Sears, 201 I; Flandro et al., 2012; Houghton et al., 2013). There also some popular science books for the layman (Allen, 1982; Smith, 1985; Tennekes, 1992; Wegner, 1997; Graig, 1997; Anderson and Eberhardt, 2001; Torenbeek and Wittenberg, 2009). Some of the later however resort to oversimplification to appeal to the reader (Sabbach, 1995). It is argued here that the main aerodynamic theory can be

Copyright C 200x lnderscience Enterprises Ltd.

An informal introduction to basic concepts of aerodynamics 3

There are many types of energies involved. Mechanical energies include potential and kinetic energies, while thermal energies include heat and internal energy (energy stored in the media and manifested by raising its temperature T).

The work done by the pressure and shear forces contribute to both mechanical and thennal energies. The sum of these six types of energy will be conserved in our analysis. According to Buckingham 1f theorem of dimensional analysis, there are at least: 15 - 4 = I I non-dimensional parameters.

There are four geometrical parameters:

Thickness ratio i = Ycav

Camber ratio Ca= Cfcav Angle of attack a( sin a= Yc)

AR= S s2 Cav Planform_area_of _wing Aspect ratio

And two from the media:

Specific heat ratio y = C~ (y = I .4 for air)

Piandtl number Pr= Cp . k

The Prandtl number is the ratio of viscosity to heat conductivity, normalised by the specific heat under constant pressure Cp.

The non-dimensional motion parameters are:

Reynolds number

Mach number

Strauhal number

p Re=-

T

v Ma=-a

The Reynolds number indicates the relative effects of viscosity, i.e.. large Reynolds number indicates small viscous effects relative to pressure (or inertia) effects.

The Mach number indicates the compressibility effects, where a is the speed of sound

(a' = : ) forisentropic process. a' -+ "' for incompressible or constant densi1y flows). The Strauhal number is a measure of unsteadiness and for steady flows

4 M. Hafez

Lift and drag can be calculated in terms of Cp and C1- There are four types of drag. The first type is friction drag; an example of that is the drag of flat plate at zero angle of attack. The second type is pressure drag as the case of flat plate normal to the flow. For three-dimensional wings, there is a vortex drag or drag due to lift and for supersonic flows, there is a wave drag. Drag calculations are complicated and they will be discussed in separate papers. In this paper, we are concerned with Jift for low speeds.

In the foJlowing, the flow is assumed steady, 2-D, incompressible, inviscid, adiabatic, and with uniform upstream conditions. These assumptions are justified in the flow region away from the airfoil surface and wake, where the viscous stresses are negligible and there is no heat transfer. It will be shown that if there is no gust, the flow is irrotational, i.e., the fluid elements have no angular velocity around their mass centres, and the motion becomes rectilinear. Such motion is much simpler to analyse than the general rotational motion.

The effects of thickness, camber ratio and angle of attack on the lift of airfoils will be also discussed.

Finally, the above discussions are limited to a continuum model where the ratio of the mean free path, l, to a characteristic length, C, is very small (i.e., Kn = ~, Knudsen

c number is much smaller than one). From kinetic theory of gases, it can be shown that:

~-Mo c Re

Flows with, M~ ~I, are called rarefied gas flow and is not covered in this study.

3 Mathematica) modeUing

9 Governing equations and boundary conditions The general case of unsteady 3-D compressible viscous flows is, of course, complicated. The governing equations are conservation of mass, momentum, and energy plus the equation of state. These are six non-linear coupled equations for six unknown functions u, v, w and the thermodynamic variables P, p, T. On the solid surface, the no slip and no penetration conditions lead to zero relative velocity between the body and the adjacent fluid. Also, the temperature T or its nonnal derivative should be specified, the later is proportional to heat flux through the boundary. In the far field, distur~ces must vanish. Even numerical solutions are not available in general. For example, transitional and turbulent flows cannot be accurately predicted.

For high Reynolds number flows at design conditions, the viscous effects are limited to boundary layers and wakes. In these cases, the above Navier Stokes equations can be replaced by Euler equations outside the viscous layers where the flow is assumed inviscid and adiabatic together with thin layer approximation in the viscous layer ignoring the second variations of the velocity and temperature in the streamwise directions compared to those in the lateral direction.

An informal introduction to basic concepts of aerodynamics 5 Further simplifications are possible most of the time. In the outer regio~ the flow can

be assumed isentropic and irrotational, allowing only for weak shocks, while the pressure variation across the boundary layer can be ignored, at least for attached flows.

Moreover, one may assume small disturbances, where the flow is almost aligned with the body, for the case of small angle of attacks, camber, and thickness ratios, allowing for simplified analysis of the outer region together with an interaction procedure to couple the inviscid flow with boundary layer calculations.

In the following, the details of some special cases of incompressible flows are discussed to understand the mystery of flight.

3.1.1 Liftingjlow over a rotating cylinder In this section a simple model of steady, inviscid, adiabatic, irrotational, incompressible, two-dimensional flow is considered to demonstrate the generation of the lift over a rotating cylinder. Both analytical and numerical solutions of the governing equations will be discussed. Two theorems of vector calculus, due to Gauss and Stokes, are needed for the following developments. Cartesian and cylindrical coordinates are used interchangeably, see Figure 1. Their relations are given by:

r2=x2+y2

tan8=% or,

x=rcos8 y=rsin8

Figure 1 Polar coordinate system

(I)

The velocity components in Cartesian coordinates are denoted by U, V and in cylindrical coordinates by U, V and related to each other, as follows,

6 MHafez

U =Ucos8-Vsin8 Or, V =Usin8+Vcos8

U =Ucos8+Vsin8 V =-Usin8+Vcos8

3.1.1.J Flow due to a source Flow due to a source at the origin is given by (see Figure 2):

u=iL.!. v=o 2irp r'

where Q is the source strength and pis the density.

Figure 2 Source flow

u

(2)

(3)

The integration of the flux over any concentric circle is the same. The above fonnula for U is valid, as long as the origin is excluded.

3.1.1.2 Flow due to a vortex A flow due to a vortex at the origin is given by (see Figure 3):

- r I -V=--, U=O 2ir r

where r is the circulation.

{4)


Figure 3 Vortex flow

The integration of the tangential velocity over any concentric circle is the same. The above formula for V is valid, as long as the origin is excluded.

The formula for the source flow is a consequence of Gauss theore~ namely the flux over a circle. including the origin, is the same as the mass flow rate generated from the source. While the formula for the 'vortex' flow is a consequence of Stokes theore~ namely the circulation over any circle, including the origin, is the same. since there is only a vortex at the origin.

3. 1. 1 .3 Gove ming equations In the control volume ab c d, there is no sources (see Figure 4). hence the flux over its boundary must vanish, i.e.,

(rpU)9 M-(rpu)N AB+(pv)E t\r-(pv)w IJ.r =0 (5) Figure 4 Control volwne for conservation of mass

y

x

8 MHafez

For incompressible flows, p is constant and it cancels out. One can divide all terms by rll.8/lr (the area of the control volume) and take the limit as Ar and ll.8 go to zero to obtain a partial differential equation:

1 ( -) 1 -- rU ,.+-Ve =0 (Sa) r r

Similarly, in the same control volume (see Figure S), there is no vortex, hence the circulation over its boundary must vanish, i.e.,

(6)

Figure S Control volume for vorticity and circulation of relation y

x

Again, one can divide by r/:18/ir and take the limit as l:!.r, l:!.8 go to zero to obtain a partial differential equation:

1 ( -) 1 -- rV ,.--Ue=O (6a) r r

Equations (Sa) and (6a) are two linear equations for U and V, (Cauchy/Reimann equations in cylindrical coordinates).

Notice, excluding the origin (r = 0), the formulas for the flow due to a source and the flow due to a vortex satisfy the governing equations (Sa) and (6a).

Equations (S) and (6) are the discrete versions, based on control volumes, and will be used later to solve the problem of finding fl and V numerically.

Uniform flow at angle of attack a is given by (see Figure 6): U = q~ cos a, V = q= sin a (7)

It can be easily seen that the above formula (7) for uniform flow satisfy the governing equations identically.

An informal introduction to basic concepts of aerodynamics 9 Figure 6 Flow at angle of attack

y

"

u x

3.1.1.4 Flows over Rankine body and Kelvin oval Consider a unifonn flow Uw in the x-direction over a source and a sink of equal strength located at equal distances along the x-direction from the origin, as shown in Figure 7.

Figure 7 Rankine body

v u

The U component is given by:

In the limit of d-+ 0, such that 2Qd remains constant, U becomes: - Dcos8 U=Ucccos8----

r2

. o Y,. I or or x (Notice: --= --- and - = - ). ox r 2 ox ax r

(8)

(8a)

Now consider a unifonn flow Ug:,, in the x-direction over two vortices rotating clockwise, located at equal distances from the origin, along the y-direction as shown in Figure 8. .

The V component is given by,

10 M Hafez

- . r 1 r 1 V=-UcosmO+ -21f (

An informal i11trod11ction to basic concepts of aerodynamics 11

Figure 9 Flow over a rotating cylinder (see online version for colours)

- - --- -

For clockwise rotation, the flow will be augmented on the top and retarded at the bottom of the cylinder, hence according to Bernoull i's law, a lift is generated proponional to r . Notice the drag is zero in this inviscid model.

For the i.ncompressiblc flows, the pressure is related to the speed as:

P 1 (-2 -2) R I - +- U +V =Const.=..!!!...+-U~ p 2 p 2 or, (12)

lnregrating the pressure forces over the rotating cylinder gives the lift and drag (per unit width):

2.z

L =- f (P-P ., )sin8Rd8 0 2z

D= - J

12 M Hafez

The circulation r can be related to the angular velocity n of the cylinder as:

r = 27rr(Qr) = u 2 (2fl)

The above relation is a consequence of Stokes theorem where the tangential velocity of the fluid particles at the surface of the cylinder is assumed to be the same as the rim velocity of the rotating cylinder. In reality, however, and due to viscous effects the lift is smaller and there is a friction drag and a pressure drag due to separation .Considering the solid body rotation of the cylinder, the circulation around the rim is equal to the area integration of the vorticity which is def med as twice the angular velocity.

Figure 1 O Control volume over a circle

3.1.1.6 Numerical solution of the governing equations Equations (5) and (6) can be solved numerically using a staggered grid as shown in Figure 11, with the boundary conditions:

U = 0 on the surface of the cylinder V from the analytical solution in the far field [equation (IO)]. (For a large domain,

one can ignore the doublet contnl>ution in the far field boundary condition). The system of the algebraic equations maybe solved via Gaussian elimination with fine meshes, the numerical solution should agree with the analytical solution everywhere.

Alternatively, iterative methods (for example, line over-relaxation) maybe used to = =

solve the central difference approximation of second order equations for U and V. where

U=rU, V=rV

and,

An informal introduction to basic concepts of aerodynamics

I(=) 1-- rU,, +-;U88 =0, r r

U = 0 on the cylinder surface and U, = _.!_ V 8 in the far field. r

I ( =) I = - rV r +2V88 =0, r r

= 1= V, =-Us =0

r on the cylinder surface, and V is given in the far field.

Figure 11 Staggered grid

+p 11

13

The same discrete algebraic equations can be obtained by manipulations of equations (5) and (6) to obtain decoupled equations for the modified velocity components, U and V.

Finally, one should remark that the analytical and numerical solutions are constructed in different ways. The analytical solution is obtained via superposition of unifonn flow plus singularities placed at the origin inside the body, with weights determined from imposing the boundary condition.

The numerical solution, on the other hand, is obtained via dividing the flow field into control volumes. To close the system, the boundary conditions at the surface of the cylinder (the no penetration condition and the far field condition are added to the field equations). The derivation of the equations is not based on superposition principle. {In fact, the control volume approach is used to derive the governing equations of compressible flow which are non-linear). Both approaches can be used to study flows around bodies other than cylinders.

Flows over certain shapes, however, can be analysed based on the development in this section (beside Rankine bodies and Kelvin ovals) using special techniques of

14 M Hafez

mapping the cylinder to other configurations. In particular, a transformation due to Jowkowski will be studied next.

3.2 Joukowski airfoils The lift generated over a rotating cylinder in a uniform stream was discovered by Magnus and an application of this was Flettner's boat. In this section, we will discuss the connection to certain airfoil shapes. Joukowski introduced a transformation to map the circle to a closed curve, with a blunt leading edge and a sharp trailing edge. Consider in general the transformation:

X=x(l+-b-2 -). Y=y(1-~) (15)

x2 + y2 x2 + y2

where d- = (b - e)2 + 2 x, y are the coordinates of the surface in the circle plane and X, Y are the coordinates

of the surface of the corresponding airfoil. e and are the coordinates of the centre of the circle, and a is its radius. For the special case of e = = 0, the circle is mapped to a slit or a flat plate, see Figure 12.

Figure 12 A circle mapped to a slit

y

-a

Notice for very large x and/or y, X=x, Y=y

y

x -2a 2a

Joukowski transformation satisfies Cauchy/Riemann equations namely,

ax_ aY = 0 and ax+ aY = 0 Ox Oy Oy Ox

x

(16)

Now, it can be shown that the flow over the circle is mapped to the flow over the airfoil. In the circle plane, the velocity components u(:c, y) and v(x, y) are governed by Cauchy/Riemann equations, together with the no penetration condition at the surface of the circle and a uniform flow in the far field. Using chain rule, u and v will be transformed to U(X, Y), V(X, Y), satisfying Cauchy/Riemann equations in the airfoil plane as we11 as the no penetration boundary condition at the airfoil surface and the same uniform flow in the far field of the X, Y plane. The details are technical and will be omitted in this discussion.


The transfonnation preserves singularities. for example. a source is transformed to a source with the same strength. Hence. the circulation around the circle is the same as the circulation around the airfoil and according to Joukowski theorem the lift is the same. since L = - P> V err.

Let's consider the circle at angle of attack a. and the corresponding flow over flat plate in the X-Y plane.

The stagnation points on the surface of the cylinder will be mapped to two points on the plate. one on the lower side and the other on the upper side. which means the flow goes around the leading and trailing edges (see Figure 13).

Figure 13 Flow over a cylinder mapped to flow over a flat plate

1 F

-a -2a

In reality. the flow leaves the trailing edge smoothly. To impose this celebrated Kutta-Joukouski conditio~ circulatory motion is added to both solutions in x, y and X, Y planes. The amount of circulation is determined such that the rear stagnation point moves to the trailing edge, hence,

r = 41rJ't'~asin(-a)

The lift coefficient of the flat plate of length 4a is then:

-n v.: r r= ) -27rsina

.!. n V2 4a 2r= For small a. sin a::: a. and CL= 27ra(a in radians).

The centre of pressure is at distance Xcp from the leading edge, and X c(c = ~.

(17)

(18)

Notice the lift on the circle is normal to the flow direction. The pressure force on the plate is, however, normal to the X-axis. The paradox is easily solved since there is a suction at the leading edge such that the net force is nonnal to the flow and equal to the lift and there is no pressure drag.

To consider the Camber effect, the circle is mapped to a circular arc by choosing e = 0, and 0 < < I (see Figure 14).

The leading and trailing edges of the circular arc are located at b while the maximum height of the arc is 2 at X = 0. (For small , the circular arc can be approximated by a parabolic arc). To apply the Kutta-Joukowski conditio~ we need again to add circulation, where

16 M Hafez

r = 4nU..,asin(-8) and,

tan8=!!.. b

For


The solution of flow over symmetric airfoil at zero angle of attack plus the solution of flat plate at angle of attack, plus the solution around a cambered arc (with zero thickness) at zero angle of attack see Figure 15. Moreover, the cambered arc can be approximated

by a parabolic arc, hence the lift coefficient is cL ::-;: 2n{a + i'). Figure 15 Flow over cambered airfoil (see online version for colours)

The centre of pressure is given by:

1 I -a+--4 2b

a+!!... b

c

+

+

(20)

In the case of Joukowski airfoils, more accurate results can be obtained by mapping the flow field from the circle plane to airfoil plane. In general, the analytical formulation is not possible and numerical methods must be used.

A final remark about generation of lift, in low speed regimes, is in place. In the case of cylinder, the source of lift is the solid body rotation, assuming that the

particles near the surface of the cylinder have the same circulation, as that of the rim velocity leading to an overestimation of the lift.

On the other hand, in the case of the airfoil, there is no rotating surface - what is the source of lift then?

One can argue that the vorticity generated in the boundary layer is the source of circulation hence the lift. Indeed, a proper application of Stokes theorem relates the circulation around the airfoil to the area integration of the vorticity in the boundary layer. Notice that the wake, in steady state case, does not contribute to circulation since the vorticity contributions cancel out.

3.3 Viscous layers

To understand viscosity effects consider steady flow between two plates, the lower plate is fixed while the other is moving with speed U.

18 MHafez

Let,P1 =P2. The only force on a small element is shear, hence

r =constant

Using Newton's viscosity law,

du r=-

dy

One obtains a linear distribution for u, (see Figure 16)

Figure 16 Flow between two plates, P 1 = P2 (see online version for colours) u=U [277 y -L 7 ~ - P2

x u=O

If P1 '/. P2, the pressure force must be considered. Far from the entrance, u is independent of x, and from conservation of mass, v = 0.

Balancing the forces yields the equation of the steady motion:

dP dr -=-

dx dy

The right hand side is a function of x, while the left hand side is a function of y, therefore both must equal a constant. Hence, r is a linear function of y and u becomes a quadratic function of y satisfying the boundary conditions.

In Figure 17, the solution is plotted for different cases, for case b, P1 is greater than P2, while for case c, P2 is greater than P1 Notice in the latter case, the flow is reversed near the fixed plate.

Figure 17 Flow between tow plates: P 1 i:P2 (see online version for colours)

c

u=O

An informal introduction to basic concepts of aerodynamics 19 This example explains flow separation over airfoils due to the retardation of an adverse pressure gradient.

The other important phenomena is transition from laminar to turbulent flow. There is a critical Reynolds number Rec, and for Re > Rec. the flow becomes unsteady and unpredictable.

For the case of U = 0 and P 1 > P2, the flow between the two plates is called fully developed (see Figure 18).

Figure 18 Fully developed flow (see online version for colours)

Near the entrance, a viscous layer is developed with an inviscid core (see Figure 19).

Figure 19 Flow near the entrance (see online version for colours)

Now, if the upper plate is removed, there will be, for high Re, a boundary layer on the lower plate as shown in Figure 20.

Figure 20 Boundaiy layer flows (see online version for colours) u u

The governing equations of conservation of mass and momentum will be complicated since both u and v become functions of x and y as the boundary layer will grow in the x-direction (see Appendix 1). Outside the boundary layer, the flow can be considered inviscid, i.e . the viscous stresses will be negligible because the variations of the velocity components are smaller. (The term inviscid is misleading since the fluid outside the boundary layer has the same viscosity coefficient as the fluid inside the boundary layer!)

20 MHafez

The numerical solution of the non-linear equations for boundary layers for case of laminar and turbulent, attached and separated flows is part of computational fluid dynamics (CFD) and it is beyond the scope of this paper.

Nevertheless, a simple case where the boundary layer does not grow, due to suction, can be analysed.

Again away from the leading edge, u is independent of x. The momentum equation in x-direction reduces to:

du dr -V-=-

dy dy

Let, ru = du , the above equation becomes: dy

dw Vru=--dy

In this case, (J) is the vorticity and it dies exponentially in they-direction.

w(y)=Ae{;y where A is a constant.

Integrating the vorticity equation, one can obtain the velocity u,

u(y)=Ae{;} +B where B = U to satisfy the far field boundary condition and A = -U to satisfy the no slip boundary condition at y = 0.

The effect of viscosity is limited to a layer next to the wall. Outside such a layer the vorticity vanishes. The thickness of this layer depends on % (see Figure 21 ). A more interesting case is considered in Appendix 2.

Figure 21 Boundmy layer with suction (see online version for colours) u=U

u

u=O

-v

The study of boundary layers is important to satisfy the no slip boundary condition and to provide an estimate for friction drag. (Ignoring boundary layers leads to D'Alembert


paradox). Coupling boundary layer calculations with inviscid flow calculations is also important to account for the pressure gradient in the streamwise direction and the possible flow separation which can be catastrophic in terms of loss of lift and increase of drag.

4 Concluding remarks

In this paper, a gentle introduction to aerodynamics is attempted. In particular, generation oflift at low speeds is explained without the use of potential theory or complex variables. Instead, the discussion is centred ~und physical variables: velocity and pressure. Appealing to Gauss and Stokes theorems, one can have a unified approach to deal with both the singularity method and the field method. In the first case, the flow is obtained due to superposition of sources and vortices inside the body and their strength can be determined by satisfying the boundary condition as in the analytical solution for the flow over a rotating cylinder. In the second approach, the flow field is decomposed into control volumes with zero sources and vorticities. Applying, Gauss and Stokes theorems leads to Cauchy/Riemann equations which in tum, can be discretised and together with the no penetration condition at solid boundaries and the far field condition, the velocity components can be calculated everywhere while the pressure is obtained from Bernoulli's law. Surface pressure integration yields the lift (and drag). (The drag for inviscid flow over closed body should be zero according to D' Alembert paradox).

In this regard, one can avoid the introduction of Cauchy/Riemann equations and apply Gauss and Stokes theorems directly at the discrete level by numerical approximation of the fluxes without dealing with partial differential equations and their discretisation. In subsequent papers, airfoil and wing theories will be discussed based on singularity methods (in terms of velocities) while numerical solutions of the flow field equations wilLbe solved (for the velocities) in particular, for non-linear problems of transonic flows (with shocks) and for boundary layers.

References Allen, J. {1982) Aerodynamics- The Science of Air in Motion, 1st ed., McGraw-Hill, New York.

(1963) by Harper and Row, New York. Anderson, D. and Eberhardt, S. (2001) Understanding Flight, McGraw-Hill, New York. Anderson, J. (1919)Jntroduction to Flight, McGraw-Hill, New York. Anderson, J. Jr. {2007) Fundamentals of Aerodynamics, McGraw-Hill, New York. Ashly, H. and Landhal, M {1965) Aerodynamics of Wings and Bodies, Dover, New York. Bertin, J. and Cmnmings, R ( 1979) Aerodynamics far Engineers, Pearson, New York. Clancy, L. (1996)Aerodynamics, Longman, New York. Dragos, L. (2003) Mathematical Methods in Aerodynamics, Kluwer, MA. Duncan, W J., Thom, AS. and Young, AD. (1972} Mechanics of Fluids, Arnold, London, UK. Flandro, G., McMahon, H. and Roach, R (2012) Basic Aerodynamics-Incompressible Flow,

Cambridge, New York. Glauert, H. (1926) The Elements of Aerofoil and Airscrew Theory, Cambridge, New York. Graig. G. ( 1997) Stop Abusing Bernoulli! How Airplanes Really Fly, Regenative Press, Indiana.

22 M Hafez Houhton. E . Carpenter. P . Collicott. S. and Valentine. D. (2013) Aerodynamics for Engineering

Students. Butterworth-Hinemann. Oxford. UK. Jones. RT. (1990) Wing Theory. Princeton, New Jersey. Karamcheti, K. ( 1966) Principles of Ideal-Fluid Aerodynamics, Wiley. New York. Katz. J. and Plotkin, A (2001) Low Speed Aerodynamics. Cambridge. New York Keuthe. A and Chow. C.Y. (1998) Foundation of Aerodynamics, Wiley. New York. Kochin, N.E . Kibel. I.A. and Roze. N.V. (1964) Theoretical Hydrodynamics, Wiley. New York Krasnov. N. (1978)Aerodynamics, NASA. Lighthill, J. (1986) An Informal Introduction to Theoretical Fluid Mechanics, Oxford. UK. Marshall, J.S. (2001)Inviscid Incompressible Flow. Wiley. New York Milne-Thomson. L.M (1958) Theoretical Aerodynamics, Dover. New York Milne-Thomson. L-M (1968) Theoretical Hydrodynamics, Dover. New York. Moran. J. (1985) Theoretical and Computational Aerodynamics. Dover. New York Munk. M. (1929) Fundamentals of Fluid Dynamics for Aircraft Designers. The Ronald Press Co,

New York Pope. A (1951) Basic Wing and Airfoil Theory, Dover. New York. Prandtl, L. and Tietjens, O.G. {1934a} Applied Hydro-Aerodynamics, Dover. New York Prandtl, L. and Tietjens, O.G. (1934b} Fundamentals of Hydro-Aerodynamics, Dover, New York. Sabbach, K. (1995) 21st Century Jet-The Making of Boeing 777, McMillan, New York Schlichting, H. (1979) Boundary Layer Theory, McGraw-Hill. New York. Sears, W. (2011) Introduction to Theoretical Aerodynamic and Hydrodynamics, AI.AA, New York. Shevell, R. (1989) Fundamentals of Flight, Prentice-Hall, New Jersey. Smetana, F. (1997) Introductory Aerodynamics and Hydrodynamics of Wings and Bodies: A

Software-Based Approach, AlAA, New York. Smith, H. {1985) The Illustrated Guide to Aerodynamics, TAB Books. PA Tennekes, H. (1992) The Simple Science of Flight, M.l.T., MA. Torenbeek, E. and Wittenberg, H. (2009) Flight Physics, Springer, New York van Dyke, M. (1964) Perturbation Methods in Fluid Mechanics, Academic Press, New York. Wegner, P. (1997) What Makes Airplane Fly?, Springer, New York White, F. (2008) Fluid Mechanics, McGraw-Hill, New York.

Appendix 1

The material in this Appendix requires some knowledge of partial differential equations, in particular Navier-Stokes equations, and it is not for a beginner.

Consider steady viscous fluid flow over a finite flat plate. The governing equations. in primitive variables, are the conservation of mass and the equations of fluid motion in the x- and y-directions, including viscous stress terms, namely,

au av -+-=0 ax ay

P Du = pu iJu + pv au = _ oP +(


PDv =puav +pvav =- oP +(o2v + 02v) Dt ax ay ax ax2 ay2

On the plate the boundary conditions are u = 0, and v = 0. The disturbances vanish in the far field, away from the plate (except in the wake).

Order of magnitude analysis Near the surface of the plate and away from the leading edge, x is of order, the length of the plate and y is of order o, the thickness of the boundary layer, where viscous stresses are of the same order as the inertia terms. For high Reynolds number flows, o

24 M Hafez

negligible and the pressure does not change much across the boundary layer. i.e . P = PJ...x) where P1 is the inviscid flow pressure at station x. Hence. the laminar boundary layer equations are:

au av -+-=0 ax ay

au au dP, a2u pu-+pv-=--+-ax ay dx ay2

with the boundary conditions u = 0, v = 0 at y = 0 and u --. u1 far away from the plate.

For the flat plate. dP, is small and can be ignored and u, = U. In more general cases. dx

the inviscid flow solution provides U = U(x) and P1 = P,(x) and in return, the boundary layer calculations provide a feedback to the inviscid flow region, for example, v at the edge of the boundary layer. Other versions of viscous/inviscid interaction procedures are available and for all cases, they replace, for high Re, the solution of Navier-Stokes equations in the whole domain. Special treatments are required however for separated flows. In the wake of the flat plate away from the trailing edge, u is governed by a linear parabolic equation wi!h exact analytical solution. For details see Schlichting (1979).

Alternative formulation An alternative formulation in tenns of velocity and vorticity is interesting and is relevant to the formulation of the inviscid flow, discussed in the main text. Navier-Stokes equations can be rewritten in the fonn:

U~+Vy =0

-Uy +V.r = (JJ and,

puf.JJ.r + pV(JJy = ( f.JJx:x + (JJYY) The first equation is conservation of mass, the second is the definition of vorticity and the third is convection/diffusion equation of the vorticity. The last equation can.be derived by eliminating the pressure from the momentum equations, differentiating the x-momentum with respect toy and they-momentum with respect to x and substracting yields the above equation after some simplification. (Or, in vector notation, taking the curl of the momentum equations eliminates the pressure tenns, since V x VP = 0 ).

Notice, the first two equations are the same as the inviscid flow equations. except the vorticity does not vanish and is obtained from the dynamics equations. Notice, also that the vorticity and its derivatives are not known at the solid surface. The vorticity vanishes in the far field, away from the body, except in the wake.

The first two equations can be combined to produce Poisson's equations for the velocity CQmponents, u and v, namely,

Uxr + U' :: -(J)y

An informal introduction 10 basic concepts of aerodynamics 25

V.a + V' = ID.r

and.

purux + pv(J)y = ( (J)x.r + (J)Y.Y) Obviously. the lrrst equation can be solved for u, the second for v and the third for (J). Since, there is no boundary condition for (J) at the solid surface. it maybe useful to solve the u- and (J)-equations in a coupled manner.

The boundary layer approximations, for high Re flows, are obvious in this case. The terms ux.r. Vx.r and (J)x.r are higher order compared to u,,,. v,,,. and (J)' respectively. The system of the reduced equations can be solved, marching in the x-direction, for attached flows (u > 0). Once again, special treatments are required for separated flows.

Notice that, it is the vorticity in the boundary layer which is responsible for the generation of circulation around the airfoil and hence the lift.

For steady flow, the rate at which vorticity is discharged into the wake, Q from upper and lower surfaces of an airfoil must be equal and opposite.

fl= 6Ju" 4Y= (u6)2 0 c3y 2

Hence, the velocity at the edge of the boundary layer must be the same for both upper and lower surfaces in the region of the trailing edge. With this condition, the circulation around any curve enclosing the boundary layer and cuts the wake at right angles to the local flow direction, is independent of the curve taken and, L = -p VJ.

The viscous/inviscid interaction procedures bring the effect of vorticity (i.e., circulation) to the inviscid flow region. Otherwise, the inviscid, irrotational flow, has zero (or any) circulation, where the flow can go around the trailing edge. The Kutta-Joukowski condition fixes the amount of circulation as discussed in the main text without the need of boundary layer calculations and its vorticity. A more physical and meaningful approach is to couple the viscous and the inviscid flow calculations with the condition that the velocity at the edge of the boundary layer is the same for both upper and lower surfaces, at the trailing edge.

Appendix 2

An interesting simple problem was introduced by Friedrichs in 1942 (see van Dyke, .1964, for more details). Consider steady viscous incompressible flow between two porous plates, a distance h apart, with suction (-V) and constant pressure gradient (dpldx). The momentum equation for the velocity component u is given by:

-V. du I 4Y =-dpl dx+ 1/Re d(du I dy)I dy The boundary conditions are: u(O) = 0 and u(h) = uo.

Let, V = I, dpldx = a, and h = 1 for convenience. If uo = I and a = 1, the exact solution is u = y. On the other hand, if a is less than 1, or uo is greater than 1, the solution will exhibit a boundary layer; its thickness is of order (I/Re). This is a consequence of requiring the viscous and inertia terms to be of the same order. Outside the boundary

26 MHafez

layer. the viscous term can be ignored and the 'inviscid solution. satisfying the boundary conditiony(l) =I. is given by:

u(y) = (1-a)+a.y In the neighbourhood of the wall, the governing equation in terms of Y = Re. y is:

-du I dY =-a/Re+d(du I dY)/ dY Ignoring the term a/Re compared to the others (for large Re). the 'viscous solution' satisfying the no slip boundary condition u{O) = o. is:

u(y) = (1-a).{l-exp(-Y)) = (l-a).(1-exp(-Re.y)) The above solution is obtained by requiring that the limit of the 'viscous solution' as Y goes to infinity, matches the limit of the 'inviscid solution' as y goes to zero. In Figure 22, the inviscid and the viscous solutions are compared with the exact solution of this model problem~ the latter is given by:

u(y,Re) = (1- a ).(1- exp(-Re.y )) I (1- exp(-Re))+ a.y

Figure 22 Comparison of viscous, inviscid, composite and exact solutions, (Re= S, a= 0.5) (see online version for colours)

I EXact ~ - - - ~ - - - ~ - lsoli~Hine)~ - -0.9 - - - .l - - - -1- - - - L - - - -' - - -I I

I I I I I I I I I I I r---7---,----r--- ---0.8 - - -1- - - -:- - - - ~ - - - ~ -- -

I I I ' I I I

I I I

, Composite 1----+---~----1-(do~ed) I

I I I I 0

7

- - - ~ - - - -:- - - - ~ -\7iscous-0.6 ___ J ___ J ____ L ___ J ____ L ___ ! ___ J __ _ I I I I I I I

I

0.5

0.4

0.3

0.2

0.1

o~:...;_-L.----1.-----L.----1.-----L-----''------'-----'-----'---~ 0 0.1 0.2 0.3 0.4 0.5 0.6 0. 7 0.8 0.9

A composite solution uc is defined as the viscous solution plus the inviscid solution minus the common part. The latter term is the limit of the viscous solution as Y goes to infinity or the limit of the inviscid solution asy goes to zero. Hence,

uc= {l-a).(1-exp(-Re.y))+a.y


It is clear that uc is a good unifonn approximation of the exact solution for large Re. In this model, the viscous solution depends on the inviscid solution and not vice versa. For typical boundary layers however, the viscous and the inviscid solutions depend on each other. Moreover, the boundary layer thickness is proportional to the inverse of the square root of Re and Re is about ftve orders of magnitude higher than that of the model problem.

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An Informal Introduction Into the Basic Concepts of Aerodynamics Part I

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