Category Theoretical and Applied Aerodynamics

A doublet is obtained when a source (Q, – a) and a sink (-Q, a) located symmetri­cally along the x-axis are merged at the origin following the limiting process

Fig. 2.5 Source: equipotental lines and streamlines

Fig. 2.6 Doublet: equipotential lines and streamlines

The results for the velocity components are

– D cos в – D sin в

Vr = , Ve =

r

where D is the strength of the doublet, D = lima^o, qaQ The results for the potential and stream functions are

The equipotential lines and streamlines are circles, tangent at the origin to the z-axis for the former and to the x-axis for the latter. See Fig. 2.6.

A line source and sink is the building block to represent profile thickness as will be seen later. If a source or a sink is placed in the plane, it is convenient to define a polar coordinate system with origin at the source or sink location. Let (r, в) represent the distance and polar angle of a point to the source/sink.

The velocity components are

Q 1

Vr = , ve = 0 (2.19)

2n r

Note that the perturbation velocity vanishes in the far field, which is consistent with the asymptotic condition. The potential and stream functions are given by

QQ

ф = ln r, f = в (2.20)

2n 2n

Q represents the source/sink intensity and has unit of volume flow rate per unit span (m2/s). Q > 0 is a source, Q < 0 a sink. The equipotential lines are circles and the streamlines are rays through the origin. See Fig.2.5.

2.2.1 Uniform Flow

The uniform flow V«, = (U, 0) is represented by the leading term of the full potential and stream functions:

Ф = Ux, Ф = Uz (2.17)

A sketch of the solution is shown in Fig.2.3.

The uniform flow = (0, W) is represented by the leading term of the full potential and stream functions:

Ф = Wz, Ф = – Wx (2.18)

A sketch of the solution is shown in Fig.2.4.

A uniform flow in an arbitrary direction can be obtained by the superposition of these two elementary solutions.

Fig. 2.3 Uniform flow: equipotential lines and streamlines

Fig. 2.4 Uniform flow: equipotential lines and streamlines

Then the irrotationality condition is identically satisfied, but substituting u and w into the conservation of mass results in

In polar coordinates this is

This is the governing equation for potential flows, the Laplace equation. The perturbation velocity field is given by grad ф — , фф, (or in short notation

Vф). The boundary conditions become

VФ. П obstacle — (UІ + Vф).П obstacle — 0 (211)

Vф — 0 as x2 + z2 — ^ (2.11)

A streamfunction can be introduced: let ty(x, z) be the perturbation streamfunc – tion. The full stream function is & — Uz + ty(x, z). The perturbation velocity components are obtained from

This is the governing equation for the streamfunction. The streamfunction also satisfies Laplace equation. The boundary conditions, however, read

& obstacle = (Uz + Ф) obstacle = const-

g■ – f) – 0 as x2 + z2 (2J6)

The first condition results from the identity V&.tobstacle — 0, where t rep­resents a unit tangent vector to the solid surface (n. t — 0), which proves that & obstacie — const., the value of this constant is however not known a priori. The

Ф = const. lines are the streamlines of the flow. Solid obstacles are streamlines and conversely, streamlines can be materialized to represent a body surface.

When both, Ф and Ф exist, simultaneously, potential and stream functions are called conjugate harmonic functions. It can be easily shown that the curves Ф = const. are orthogonal to the curves Ф = const.

The boundary conditions determine the solution of the PDEs. They correspond to the necessary and sufficient conditions for the well-posedness of solutions to the PDEs, i. e. to exist, be unique and depend continuously on the data. There are two types of boundary conditions traditionally associated with the above system, the tangency condition and the asymptotic condition in the far field.

The tangency condition expresses the fact that in an inviscid flow, the fluid is expected to slip along a solid, impermeable surface. This translates to

(V. n)obstade = Vn = 0 (2.4)

where Vn represents the normal relative velocity with respect to the solid surface.

Other conditions can be prescribed, for example in the case of a porous surface, the normal relative velocity can be given as

Vn = g (2.5)

where g is a given function along the surface.

The asymptotic condition states that, for a finite obstacle, the flow far away from the body, returns to the uniform, undisturbed flow V«,. In terms of u and w this reads

u, w ^ 0 as x2 + z2 (2.6)

2.1 Background

2.1.1 Notation

In two-dimensions, the obstacles are cylinders of various cross sections, whose axis is perpendicular to the plane in which the flow is taking place (Fig. 2.1). When a wing will be considered, the wing span will be described by the variable y. Hence, the flow past cylinders will take place in the (x, z) plane, see Fig. 2.2. In this chapter we will consider steady, 2-D, inviscid, incompressible, adiabatic and irrotational flow, also called potential flow. The influence of gravity will be neglected.

2.1.2 Governing Equations

Lets (u, w) represent the perturbation velocity or deviation from the uniform flow V«, = (U, 0). At any point in the flow field the velocity is V = (U + u, w), the density is p = const. and the pressure p. The conservation of mass and irrotationality condition form a system of two linear first-order partial differential equations (PDEs) for (u, w) as

(2.1)

In local polar coordinates (r, в), the system reads in terms of V = (Vr, Ve)

dr Vr і dve 0

dr + de =v

dr ve dvr 0

dr дв = 0

© Springer Science+Business Media Dordrecht 2015

J. J. Chattot and M. M. Hafez, Theoretical and Applied Aerodynamics,

DOI 10.1007/978-94-017-9825-9_2

Fig. 2.2 Coordinate system and notation

where Vr = U cos в + vr and Ve = —U sin в + ve are the total velocity components, including the contribution of the uniform flow. The irrotationality condition is the consequence of conservation of momentum for inviscid, adiabatic, incompressible flow. In other words, a solution to the above system also satisfies the momentum equations. The energy equation (conservation of mechanical energy) is also satisfied, under the stated assumptions.

The pressure is obtained from Bernoulli’s equation, again a consequence of the momentum conservation (it can be interpreted also as a conservation of mechanical energy per unit mass, where the potential energy is constant):

l + 2 V2 = ^ + 1 V^= 1U2 = const.

p 2 p 2 p 2

Note that the Bernoulli’s equation is nonlinear. However, as we will see, it can be linearized under the assumption of small perturbation.

In the text, the theory is developed for different regimes and the numerics are explained for some problems. In this respect, some computer code are available for full potential and transonic small disturbance equations. Boundary layer codes for laminar and turbulent flows are also available.

In Fig. 1.6, a road map for the materials of the book is described. The block in the middle covers steady two-dimensional incompressible inviscid flows. The generation of lift around a rotating cylinder (Magnus effect) is mathematically modeled as

a potential flow with circulation. In the text, the important notion of circulation is discussed in details from both physical and mathematical views. Without circulation, there is no lift (because of symmetry). The circulation increases the velocity on one side and retards it on the other side. The difference in velocity is associated with a difference in pressure (Bernoulli’s law), hence a lifting force is generated. An analytical closed form solution is obtained for this model. The lift is related to the asymmetry generated by the circulation and a relation between lift and the angle of the flow separation is derived. Also, the Kutta-Joukowski lift theorem is established.

Next, the Joukowski transformation is introduced to relate the flow over a cylinder to a flow over Joukowski airfoils (including the flat plate and circular arc as special cases). The celebrated formula for the lift coefficient of flat plates (or for that matter any thin airfoil) is derived. Similarly, a formula for the camber effect is obtained from the circular arc. In all cases, the Kutta-Joukowski condition at the sharp trailing edge is applied, which states that the flow leaves the airfoil smoothly to mimic the real viscous flow. The physical source of circulation (and hence lift) can be explained in terms of the vorticity of the fluid elements in the boundary layer, due to shear stresses between the layers of the fluid. The general Joukowski airfoil results are also obtained (without using complex variables) and the effect of thickness on lift is delineated.

Next, the thin airfoil theory of Glauert and Munk is discussed in details, allowing the calculation of lift (and moment) for general thin airfoils using superposition of fundamental solutions (in terms of sources for the thickness problems and vortices for the lifting problems). The above topics are covered in Chaps. 2 and 3.

Compressibility effects for two-dimensional subsonic, transonic and supersonic flows are discussed in Chap.4. Both the linearized theory and the nonlinear transonic small disturbance theory are treated in detail. For subsonic flows, the Prandtl-Glauert rule relating subsonic and incompressible flow results is derived. For supersonic flows, Ackeret theory for thin airfoils provides the lift and wave drag estimates, as well as an explanation for why we have this type of drag. For transonic flows, the failure of the linearized theories is explained and the characteristic and shock relations, based on small disturbance theory, are given. Numerical schemes to solve the nonlinear equation using the Murman-Cole method are explained and the numerical results are presented. This chapter is one of the unique features of this book.

Chapter 5 deals with two-dimensional unsteady cases for both incompressible and compressible flows.

Three-dimensional effects are studied in Chaps. 6 and 7 for high and low aspect ratio (AR) wings. For incompressible flows, Prandtl lifting line theory (1918), the Weissinger vortex lattice method and the Munk and Jones slender wing theory are covered. The concept of induced drag (or vortex drag), also called drag due to lift, is explained and it is shown that for given lift, an elliptic planform (without twist) is optimum in the sense that the induced drag is minimum (the induced drag is related to the added kinetic energy stored in the cross flow induced by the vortex sheet, that is equal to the thrust needed to maintain the wing motion). The extension to compressible flows, with low and high Mach numbers, is followed for both bodies of revolution and slender wings. Optimum shapes for supersonic projectiles are discussed, together with the transonic and supersonic area rules, as well as conical flows. Transonic lifting line theory, swept and oblique wings are also covered.

Finally, viscous effects or Reynolds number effects, are dealt with in Chaps. 8 and 9, starting with Navier-Stokes equations and incompressible and compressible boundary layer theory, including viscous/inviscid interaction procedures, and the calculation of skin friction drag and form drag (that is pressure drag due to the boundary layer interaction with the inviscid flow, which is important in the case of separation. In general, there are four types of drag, skin friction drag, induced drag, wave drag and form drag. It is difficult to separate their contributions, particularly if the shock penetrates the boundary layer).

The book contains also four special topics, including wind turbines, airplane design projects, hypersonic flows and flow analogies (electric and hydraulic). Another special feature of the book is three appendices, the first two consist of ten exams and their solutions, and the last deals with mathematical methods in aerodynam­ics, including the method of complex variables, method of characteristics and the conservation laws with shock waves and contact discontinuities.

There are already many text books available on aerodynamics (see Refs. [1-12] below) and the authors would like to add one more. The authors hope that the reader will enjoy this book as much as the authors did.

In general, modeling of fluid motion is a complicated subject particularly with high speed turbulent flows. It is argued that simple models based on small disturbance potential flow equation, with boundary layer corrections are attractive, at least for educational purposes, to identify the difficulties of relevant phenomena at design conditions and to study successfully practical solutions for these problems.

The students should be aware of the limitations of these simple models and the range of their validity. In the following chart, different formulations are listed with their relations, Fig. 1.5.

In the common commercial and military flight regimes, continuum mechanics assumptions are acceptable and the Navier-Stokes equations are considered the high fidelity model. For high Reynolds number flows, the viscous stresses and the heat conduction process are ignored outside a thin layer adjacent to the solid surfaces and in the wake.

Assuming inviscid and adiabatic flows, the Navier-Stokes equations reduce to the Euler equations, representing conservation of mass, momentum and energy for a perfect gas.

Far field: no disturbances (except 2—D)

irrotational

B. C. at solid surface: no slip, no penetration, temperature or heat transfer prescribed Fig. 1.5 From Navier-Stokes to linearized models

Adjacent to the solid surface, the thin layer approximation is based on ignoring the second order tangential derivatives in viscous and heat conduction terms, compared to the corresponding derivatives normal to the flow direction. Notice that the Euler equations are a subset of these equations. Therefore, outside a thin layer (its size depends on Reynolds number), all the viscous and heat conduction terms are ignored.

The Euler equations admit several modes, namely acoustic, entropy and vorticity modes (vorticity is twice the angular velocity of the fluid element around its center). For steady flows, with uniform upstream conditions, entropy is generated across shock waves, and for curved shocks, vorticity is generated due to variation of entropy from one streamline to another, according to a famous relation due to Crocco. At design conditions, shocks are usually assumed weak, hence isentropic conditions can be used. It follows then that vorticity vanishes. Irrotational flows are easier to handle since the fluid particles do not rotate around their centers as in rotational flows. With the assumption of zero vorticity, a potential function can be introduced (thanks to Stokes’ theorem), such that the velocity components are represented in terms of the partial derivatives of the potential function. Moreover, an integral of motion exists. For steady, isentropic, irrotational flows, the momentum equations can be integrated to give the most celebrated equation in fluid mechanics, namely Bernoulli’s law, which provides a relation for the pressure (or density) in terms of the velocity magnitude. Hence, the governing equations are reduced to one (nonlinear) equation for the potential function, in terms of one parameter, i. e. the free stream Mach number.

The steady potential flow solution (usually) does not satisfy the no slip boundary condition at a solid surface, since viscous effects are ignored, leading to zero skin friction drag (d’Alembert paradox). Prandtl introduced the boundary layer concept in 1904, to rectify these deficiencies. At least for attached flows over smooth streamlined thin bodies, the velocity in the boundary layer is almost tangent to the body surface, i. e. the velocity in the normal direction is small and its variation in this direction is also small, hence the pressure gradient normal to the wall is negligible.

Together with the thin layer approximation, boundary layer theory provides a pow­erful tool to fix the potential flow formulation, provided, of course, a viscous/inviscid interaction procedure is efficiently implemented. The boundary layer flow pushes up the flow outside the boundary layer and thus a pressure distribution different from the one without the boundary layer is established. This pressure distribution in return affects the boundary layer flow and the coupling is important to obtain the right interaction.

Both potential flow and Prandtl boundary layer models can be further simplified for thin bodies with attached flows. The full potential equation can be reduced to a small disturbance transonic equation, which is still nonlinear, and is able to model flows with both local subsonic and local supersonic regions separated by sonic line and possibly weak shocks. Assuming the flow profile inside the boundary layer, one can obtain a simplified formulation for the governing equations there following von Karman’s analysis. With these models, the coupling procedures are also simplified.

Finally, to obtain analytical solutions, at least for pure subsonic and pure super­sonic flows, the nonlinear small disturbance equation is linearized. Ignoring the

nonlinear terms can be justified away from the transonic regime. Superposition tech­niques can be used to establish solutions for realistic shapes. Also, empirical formulae can be used to solve the von Karman equation, in the boundary layer, for both laminar and turbulent flows.

In Fig. 1.5, the boundary conditions at the solid surface are the no slip and no penetration conditions for the velocity together with the specification of wall tem­perature or heat flux. In the far field, disturbances die out (except in some two dimensional cases).

Aerodynamics is the study of the forces on a body in a relative motion with air. Aerodynamics is a subset of fluid mechanics. Other related subjects are hydrody­namics and gas dynamics where the medium is water or gas (at high temperature). In this book we are mainly interested in the aerodynamics of airplanes, also rockets, propellers and wind mills, Fig. 1.1.

Other applications include aerodynamics of cars, trains, ships, sails, buildings and bridges. Aerodynamics applications in technology are of course in turbomachinery (compressors and turbines) and in heating, drying and mixing processes (for example aerodynamics of combustion).

Aerodynamics in nature is manifested in flying birds, also in atmospheric bound­ary layers, storms, tornadoes and hurricanes and their effects on objects like trees as well as surface erosion and land desertification.

There are three ingredients in any aerodynamic analysis: the geometry of the con­figuration or the model, the medium or the fluid, and the relative motion or the flow.

Consider an airplane flying in the air and in particular the wing. The geometry of a typical wing can be characterized by the maximum camber d and maximum thickness e of a cross section (airfoil) with maximum chord c and span b, see Fig. 1.2.

The medium (air) can be characterized by certain properties: viscosity coefficient (p), coefficient of heat conductivity (k) and the specific heats cp and cv under constant pressure and constant volume respectively.

For steady flight, as in cruising speed conditions, the flow features including the forces on the body are the same as those obtained if the wing is fixed and air is blown on it. What counts is the relative motion between the body and the particles of the air. Figure1.3 demonstrates the two cases. This observation is the basis of wind tunnel testing. It is easier for analytical and numerical analysis to fix the body and solve the steady state equations in this frame where the variables do not change with time!

The flow can be described in terms of the local velocity vector (V) and the thermodynamic variables: pressure (p), friction (t), density (p) temperature (T)

© Springer Science+Business Media Dordrecht 2015 3

J. J. Chattot and M. M. Hafez, Theoretical and Applied Aerodynamics,

DOI 10.1007/978-94-017-9825-9_1

and pulsation (w) when oscillations are present. The velocity has magnitude and direction. Again what counts is the relative angle between the body axis and the wind (i. e. the same results are obtained if the wind is in the horizontal direction and the wing at angle of attack or the wing is horizontal and the wind inclined with the same angle as shown in Fig. 1.4). In the case of a cambered airfoil, the angle of attack and the camber ratio are the parameters indicative of the flow asymmetry.

Fig. 1.4 Wing at angle of attack

In Tablel.1, the above 13 (or 14) dimensional quantities are summarized together with the corresponding nondimensional parameters. The nondimensional parameters are identified once reference quantities are chosen. For example, the relative camber d, the thickness ratio e and the aspect ratio AR = S a b where S a bc is the area of the wing. The incoming flow velocity U is chosen as reference velocity. The ratio of the specific heats y = cp and the angle of attack a are obviously nondimensional. On the other hand the viscosity and heat conductivity coefficients have different units. A nondimensional parameter of the ratio of viscosity to heat conductivity is introduced as Pr = pr and is called the Prandtl number.

The ratio of the speed of the body (or the wind) U to the speed of sound a is called the Mach number, M = U (where a = VyRT = for a perfect gas). The Mach number represents the compressibility effects.

The ratio of the normal stress (pressure p) to the tangential stress (friction т) is related to the Reynolds number Re a —. The conventional definition is Re = PU. The Reynolds number represents the ratio of inertia to viscous stress as well. The flow oscillations are characterized by the Strouhal number St = Ц-. The result for pressure or shear stress depends on the parameters via a dimensional relationship of the form

p (or т) = f (b, c, d, e, cp, cv, k, p, T, U, p, w, a)

The study of aerodynamics is mainly to find the nondimensional stresses:

• the pressure coefficient Cp = prpr^f

• the skin friction coefficient Cf = – rL

2 pU

as functions of the nondimensional parameters of flight representing the geometry, the medium and the motion, i. e.

Notice in the definition of Cp, indicates the atmospheric pressure and 1 pU2 is called the dynamic pressure. (Cp should not be confused with cp, the latter being the specific heat under constant pressure.)

The relation given above is consistent with the Buckingham PI theorem. Since we have four basic units (length, time, mass and temperature) and 13 (or 14) dimensional quantities, hence there are at least 9 (or 10) nondimensional parameters, in fact 10 (or 11) since a is already a dimensionless parameter.

Integration of surface stresses provides the moment about a point (or an axis) and the forces, the lift, and the drag (the force normal and in the flow direction respectively).

Our study is limited to small angles of attack, relative cambers and thickness ratios, i. e. |a| ^ 1, |d | ^ 1 and | ^ 1. Moreover, 7 and Pr are constants. For air, under normal conditions, 7 = 1.4 and Pr = 0.72.

Three types of wings are of interest: high and low aspect ratio wings and their limits, two dimensional and axisymmetric bodies, as well as wings of aspect ratio of order one.

The Mach number range includes constant density, incompressible flow (M = 0 not because V is zero, but because a is large) and compressible subsonic M < 1, transonic M ~ 1 and supersonic flows M > 1. A short introduction of hypersonic flows, M > 1 is presented but rarefied gas dynamics is not included in this book.

Only high Reynolds number flows are of interest in conventional aerodynamics including laminar, transitional and turbulent flows, depending on critical values of Reynolds number.

The purpose of this book is to expose students to the classical theories of aero­dynamics to enable them to apply the results to a wide range of projects, from aircraft to wind turbines and propellers. Most of the tools are analytical, but computer codes are also available and are used by the students to carry out seven to eight projects during the course of a quarter. These computer tools can be found at http://mae. ucdavis. edu/chattot/EAE127/ along with the project statements.

The main focus is on aircraft and the theories and codes that help in estimating the forces and moments acting on profiles, wings, wing-tail and fuselage configu­rations, appropriate to the flow regime, i. e., subsonic, transonic, supersonic, viscous or inviscid, depending on the Mach number and Reynolds number.

The book culminates with a study of the longitudinal equilibrium of a glider and its static stability, a topic that is not usually found in an aerodynamics but in a stability and controls book. This chapter reflects the expertise of one of the authors (JJC), who has been involved for several years in the SAE Aero Design West competition, as faculty advisor for a student team, (http://students. sae. org/ competitions/aerodesign/west/) and has developed the tools and capabilities enabling students to develop their own designs and perform well in the competition. As all airplane modelers know, placing the center of gravity in the correct location is critical to the viability of an aircraft, and a statically stable remote controlled model is a requirement for human piloting.

The material is presented in a progressive way, starting with plane, two­dimensional flow past cylinders of various cross sections and then by mid-quarter, moving to three-dimensional flows past finite wings and slender bodies. In a similar fashion, inviscid incompressible flow is followed by compressible flow and tran­sonic flow, the latter requiring the numerical solution of the nonlinear transonic small disturbance equation (TSD). Viscous effects are discussed and also, due to nonlinear governing equations, numerical simulation is emphasized.

A set of problems with solutions is placed in Part III. It corresponds to final examinations given over the last 10 years or so that the students have 2 hours to complete.

Finally, the reader is assumed to have the basic knowledge in fluid mechanics that can be found in standard textbooks on this topic, in particular as concerns the physical properties of fluids (density, pressure, temperature, equation of state, viscosity, etc.) and the conservation theorems using control volumes. The reader is also assumed to master undergraduate mathematics (calculus of single variables, vector calculus, linear algebra, and differential equations). Three appendices are included in the book, summarizing the material relevant to the subject of interest.

Aerodynamics has a long history and it has reached a mature status during the last century. There are at least 20 books written on aerodynamics in the last 20 years (see references). Some of these are excellent textbooks and some are outdated or out of print. All of the existing texts are based, however, on small disturbance theories. These theories are essential to gain understanding of the physical phenomena involved and the corresponding structure of the flow fields. They also provide good approximations for some simple cases. For practical problems, however, there is a demand for accurate solutions using modern computer simulation. Small distur­bance theories can still provide special solutions to test the computer codes. More important perhaps, they can provide a guideline to construct accurate and efficient algorithms for practical flow simulations. They are also used to develop the far field behavior required for the numerical solution of the boundary value problems. In general, the linearized boundary conditions and the restriction to Cartesian grids are no longer sufficient. Grid generation algorithms for complete airplanes, although still a major task in a simulation, are nowadays used routinely in industry. Hence, small disturbance approximations are no longer necessary and indeed full nonlinear potential flow codes, developed over the last two decades, are available everywhere. While it is argued that the corrections to potential flow solutions due to vorticity generated at the shocks can be ignored for cruising speed at design conditions, the viscous effects are definitely important to assess. Again, boundary layer approxi­mations can be useful as a guideline to construct effective viscous/inviscid inter­action procedures.

In the book we adopt this view in contrast to a complete CFD approach based on the solution of the Navier-Stokes equations everywhere in the field for more than one reason: it is more attractive, from an educational viewpoint, to use potential flow model and viscous correction. It is also more practical, since Euler and hence Navier-Stokes codes are more expensive and subject to errors due to artificial viscosity as a result of the discrete approximations. A simple example is the accurate capturing of the wake of a wing and the calculation of induced drag, still a challenge today; for the same reasons, the simulations of propellers and helicopter rotor flows are in continuing development, let alone, the problem of turbulence.

In the text, the formulation and the numerics are developed progressively to allow for both small disturbances and full nonlinear potential flows with viscous/ inviscid interactions. Only a few existing books (two or three) address these issues and we hope to cover this material in a thorough and simple manner.

The book contains an extensive list of references on aerodynamics Including textbooks, advanced and specialized books, classical and old books, flight mechanics books as well as references cited in the text.

Davis, California J. J. Chattot

November 2013 M. M. Hafez