Aerodynamics Lecture 1: Introduction - Equations of Motion G. Dimitriadis
Definition Aerodynamics is the science that analyses the flow of air around solid bodies The basis of aerodynamics is fluid dynamics Aerodynamics only came of age after the first aircraft flight by the Wright brothers The primary driver of aerodynamics progress is aerospace and more particularly aeronautics
Applications (1) Basic phenomena: Flow around a cylinder Shock wave Flow around an airfoil
Applications (2) Low speed aerodynamics Trailing vortices High lift devices
Applications (3) High speed aerodynamics F14 shock wave causes condensation F14 shock wave visualized on water s surface
New concepts: Blended wing body Micro-air vehicles Forward-swept wings Applications (4)
Applications (5) Space: Rockets, spaceplanes, reentry, Airship 1
Applications (6) Non-aerospace applications: cars, buildings, birds, insects
Categories of aerodynamics Aerodynamics is an all-encompassing term It is usually sub-divided according to the speed of the flow regime under investigation: Subsonic aerodynamics: The flow is subsonic over the entire body Transonic aerodynamics: The flow is sonic or supersonic over some parts of the body but subsonic over other parts Supersonic aerodynamics: The flow is supersonic over all of the body Hypersonic aerodynamics: The flow is faster than four times the speed of sound over all of the body
Flow type applications Subsonic aerodynamics: Low speed aircraft, high-speed aircraft flying at low speeds, wind turbines, environmental flows etc Transonic aerodynamics: Aircraft flying at nearly the speed of sound, helicopter rotor blades, turbine engine blades etc Supersonic aerodynamics: Aircraft flying at supersonic speeds, turbine engine blades etc Hypersonic aerodynamics: Atmospheric re-entry vehicles, experimental hypersonic aircraft, bullets, ballistic missiles, space launch vehicles etc
Content of this course (1) This course will address mostly subsonic and supersonic aerodynamics Transonic aerodynamics is very difficult and highly nonlinear Small perturbation linearized solutions exist but their accuracy is debatable Hypersonic aerodynamics is beyond the scope of this course
Content of this course (2) Subsonic aerodynamics Incompressible aerodynamics Ideal flow 2D flow 3D flow Viscous flow Viscous-inviscid matching Compressibility corrections Supersonic aerodynamics 2D flow 3D flow
Simplifications The different categories of aerodynamics exist because of the different amount of simplifications that can be applied to particular flows Air molecules always obey the same laws, irrespective of the size or speed of the object that is passing through them However, the way we analyze flows changes with flow regime because we apply simplifications Without simplifications very few useful results can be obtained
Full Navier Stokes Equations The most complete model we have of the flow of air is the Navier Stokes equations These equations are nevertheless a model: they are not the physical truth They represent three conservation laws: mass, momentum and energy They are not the physical truth because they involve a number of statistical quantities such as viscosity and density
t + u t u t v t w t E + u2 Navier Stokes for Aerodynamicists + v + w y z + uv + uw + ue + uv y + v 2 y + vw y + ve y + u xx + v xy + w xz = 0 + uw z + vw z + w 2 z + we z = xx + xy y + xz z = xy + yy y + yz z = xz + yz y + zz z = q t + uq + vq y + wq z + ( y u xy + v yy + w yz ) + ( z u xz + v yz + w zz )
Nomenclature The lengths x, y, z are used to define position with respect to a global frame of reference, while time is defined by t. u, v, w are the local airspeeds. They are functions of position and time. p,, are the pressure, density and viscosity of the fluid and they are functions of position and time E is the total energy in the flow. q is the external heat flux
The stress tensor Consider a small fluid element. In a general flow, each face of the element experiences normal stresses and shear stresses The three normal and six shear stress components make up the stress tensor
More nomenclature The components of the stress tensor: xx = p + 2μ u, yy = p + 2μ v y, zz = p + 2μ w z xy = yx = μ v + u, y yz = zy = μ w y + v, z zx = xz = μ u z + w The total energy E is given by: E = e + 1 ( 2 u2 + v 2 + w 2 ) where e is the internal energy of the flow and depends on the temperature and volume.
Gas properties Do not forget that gases are also governed by the state equation: p = RT Where T is the temperature and R is Blotzmann s constant. For a calorically perfect gas: e=c v T, where c v is the specific heat at constant volume.
Comments on Navier-Stokes equations Notice that aerodynamicists always include the mass and energy equations in the Navier- Stokes equations Notice also that compressibility is always allowed for, unless specifically ignored This is the most complete form of the airflow equations, although turbulence has not been explicitly defined Explicit definition of turbulence further complicates the equations by introducing new unknowns, the Reynolds stresses.
Constant viscosity Under the assumption that the fluid has constant viscosity, the momentum equations can be written as u t v t t w + u2 + uv + uw + uv y + v 2 y + vw y + uw z + vw z + w 2 z = p + μ 2 u + 2 u 2 y + 2 u 2 z 2 = p y + μ 2 v + 2 v 2 y + 2 v 2 z 2 = p z + μ 2 w 2 + 2 w y 2 + 2 w z 2
Compact expressions There are several compact expressions for the Navier-Stokes equations: Tensor notation: Du i Dt p = + μ 2 u i 2 i i Vector notation: u t + 1 2 u u + ( u) u = p + μ2 u Matrix notation: u t + T uu T T = p + μ2 u
Non-dimensional form The momentum equations can also be written in non-dimensional form as u t v t t w + u2 + uv where + uw + uv y + v 2 y + vw y + uw z + vw z + w 2 z = p + = p y + = p z + 1 2 u Re + 2 u 2 y + 2 u 2 z 2 1 2 v Re + 2 v 2 y + 2 v 2 z 2 1 2 w Re 2 + 2 w y 2 + 2 w z 2 =, u = u, v = v, w = w, x = x U U U L, y = y L, z = z L, t = tl U, p = p 2 U
Solvability of the Navier- Stokes equations There exist no solutions of the complete Navier-Stokes equations The equations are: Unsteady Nonlinear Viscous Compressible The major problem is the nonlinearity
Flow unsteadiness Flow unsteadiness in the real world arises from two possible phenomena: The solid body accelerates There are areas of separated flows This course will only consider solid bodies that do not accelerate Attached flows will generally be considered Therefore, unsteady terms will be neglected All time derivatives in the Navier-Stokes equations are equal to zero
Unsteadiness Examples Flow past a circular cylinder visualized in a water tunnel. The airspeed is accelerating. The flow is always separated and unsteady. It becomes steadier at high airspeeds Flow past an airfoil visualized in a water tunnel. The angle of attack is increasing. The flow attached and steady at low angles of attack and vice versa.
Viscosity Viscosity is a property of fluids All fluids are viscous to different degrees However, there are some aerodynamic flow cases where viscosity can be modeled in a simplified manner In those cases, all viscous terms are neglected.
Cases where viscosity is Shock wave important Boundary layer Wake
Euler equations Neglecting the viscous terms, we obtain the unsteady Euler equations: t + u t u t v t w t E + u2 + v + w y z + uv + uw + ue + uv y + v 2 y + vw y + ve y = 0 + uw z + vw z + w 2 z + we z = p = p y = p z up = ( vp ) y wp z
Classic form of the Euler equations The Euler equations are usually written in the form: where U t + F + G y + H z = 0 u v w u p + u 2 uv uw U = v, F = uv, G = p + v 2, H = vw w uw uw p + w 2 E u( E + p ) v( E + p ) w( E + p )
Steady Euler Equations Neglecting unsteady terms we obtain the steady Euler equations: u u 2 uv uw + v y + uv y + v 2 y + vw y + w z = 0 + uw z + vw z + w 2 z = p = p y = p z
Example 1 Notice that in the steady Euler equations, the energy equation has disappeared. Show that neglecting unsteady and viscous terms turns the energy equation into an identity if the air s internal energy is constant in space.
Compressibility The compressibility of most liquids is negligible for the forces encountered in engineering applications. Many fluid dynamicists always write the Navier- Stokes equations in incompressible form. This cannot be done for gases, as they are very compressible. However, for low enough airspeeds, the compressibility of gases also becomes negligible. In this case, compressibility can be ignored.
Compressibility examples Hypersonic flow over blunt wedge Transonic flow over airfoil Supersonic flow over sharp wedge
Incompressible, steady Euler Equations The incompressible, steady Euler equations become u + v y + w z = 0 u u + v u y + w u z = 1 p u v + v v y + w v z = 1 p y u w + v w y + w w z = 1 p z
Comment on the Euler equations The Euler equations are much more solvable than the Navier-Stokes equations They are most commonly solved using numerical methods, such as finite differences There are very few analytical solutions of the Euler equations and they are not particularly useful In order to obtain analytical solutions, the equations must be simplified even further
Rotational flow: Flow rotationality Fluid rotation Fluid particle, time t 1 Irrotational flow: No fluid rotation Fluid particle, time t 2 Fluid particle, time t 3 Fluid particle, time t 1 Fluid particle, time t 2 Fluid particle, time t 3
Irrotationality (1) Some flows can be idealized as irrotational In general, attached, incompressible, inviscid flows are also irrotational Irrotationality requires that the curl of the local velocity vector vanishes: u = 0 where u=ui+vj+wk and = i + y j + z k
Irrotationality (2) This leads to the simultaneous equations: w y v z = 0, w u z = 0, v u y = 0 Integrating the momentum equations using these conditions leads to the wellknown Bernoulli equation 1 2 u2 + v 2 + w 2 + P = constant
Example 2 Integrate the incompressible, steady momentum equations to obtain Bernoulli s equation for irrotational flow You can start with the 2D equations
Velocity potential Irrotationality allows the definition of the velocity potential, such that It can be seen that all three irrotationality conditions are satisfied by this function Substituting these definitions in the mass equation leads to u = -, v = - y, w = - z 2 2 + 2 y 2 + 2 z 2 = 0
Laplace s equation The irrotational form of the Euler equations is Laplace s equation. This is an equation that has many analytical solutions. It is the basis of most subsonic, attached flow aerodynamic assumptions. The equation is linear, therefore its solutions can be superimposed The complete flow problem has been reduced to a single, linear partial differential equation with a single unknown, the velocity potential.
Potential flow Incompressible, inviscid and irrotational flow is also called potential flow because it is fully described by the velocity potential. The first part of this course will look at potential flow solutions: First in two dimensions Then in three dimensions Potential flow solutions have provided us with the most useful and trustworthy aerodynamic results we have to date. Their limitations must be kept in mind at all times.
Potential flow solutions We now have a basis for modelling the flow over 2D or 3D bodies. All we need to do is: Solve Laplace s equation With two boundary conditions (2 nd order problem): Impermeability: Flow cannot enter or exit a solid body Far field: The flow far from the body is undisturbed.
Boundary conditions (1) Neumann boundary condition n: unit vector normal to the surface q n : normal flow velocity component q t : tangential flow velocity component Impermeability: The normal flow velocity component must be equal to zero. q n = n surface =0 q n n qt
Boundary conditions (1bis) Dirichlet boundary condition An alternative form of the impermeability condition states that the potential inside the body must be a constant: (x,y,z) i (x,y,z) i (x,y,z)=constant
Boundary conditions (2) Far field: Flow far from the body is undisturbed. This usually is expressed as: * 0, as r r r 2 =x 2 +y 2 +z 2 r
2D Potential Flow Two-dimensional flows don t exist in reality but they are a useful simplification Two-dimensionality implies that the body being investigated: Has an infinite span Does not vary geometrically with spanwise position As examples, consider an infinitely long circular cylinder or an infinitely long rectangular wing
2D Potential equations Laplace s equation in two dimensions is simply 2 + 2 2 y = 0 2 While the irrotationality condition is v u y = 0 We still need to find solutions to this equation.
Streamlines A streamline is a curve that is instantaneously tangent to the velocity vector of the flow x is the position vector of a point on a streamline, u is the velocity vector at that point and s is the distance on the streamline of the point from the origin x s u
Streamline definition A streamline is defined mathematically as: dx ds = u Where u has components u, v, w and x has components x, y, z. It can be easily seen that the definition leads to: dx ds = u, dy ds dz = v, ds = w, and therefore dx u = dy v = dz w
The stream function The stream function is defined at right angles to the flow plane, i.e. u = Where u=[u v 0] and =[0 0 ]. It can be seen that u = y, v = - The stream function is only defined for 2D or axisymmetric flows.
Properties of the stream function The stream function automatically satisfies the continuity equation. u + v y = + y y = 2 y 2 y = 0 The stream is constant on a flow streamline d = But, on a streamline Therefore dx + y dy = vdx + udy dx u = dy v d = udy + udy = 0
Elementary solutions There are several elementary solutions of Laplace s equation: The free stream: rectilinear motion of the airflow The source: a singularity that creates a radial velocity field around it The sink: the opposite of a source The doublet: a combined source and sink The vortex: a singularity that creates a circular velocity field around it.
Historical perspective 1738: Daniel Bernoulli developed Bernoulli s principle, which leads to Bernoulli s equation. 1740: Jean le Rond d'alembert studied inviscid, incompressible flow and formulated his paradox. 1755: Leonhard Euler derived the Euler equations. 1743: Alexis Clairaut first suggested the idea of a scalar potential. 1783: Pierre-Simon Laplace generalized the idea of the scalar potential and showed that all potential functions satisfy the same equation: Laplace s equation. 1822: Louis Marie Henri Navier first derived the Navier-Stokes equations from a molecular standpoint. 1828: Augustin Louis Cauchy also derived the Navier-Stokes equations 1829: Siméon Denis Poisson also derived the Navier-Stokes equations 1843: Adhémar Jean Claude Barré de Saint-Venant derived the Navier- Stokes equations for both laminar and turbulent flow. He also was the first to realize the importance of the coefficient of viscosity. 1845: George Gabriel Stokes published one more derivation of the Navier- Stokes equations.