AerodynamicsME-438

Spring’16ME@DSU

Dr. Bilal A. Siddiqui

Review of Vector Calculus• Gradient of a scalar is a vector

• Curl of a vector is a vector

• Divergence of a vector is a scalar

Review of Vector Calculus• Let A be a vector and p be a scalar. The line, surface and volume

integrals are related by

Review of Fluid Mechanics

Review of Fluid Mechanics

Review of Fluid Mechanics

Review of Fluid Mechanics

Review of Fluid Mechanics

Review of Fluid Mechanics

Note: We have made no assumption of inviscidity or compressibility

Review of Fluid MechanicsNote: We have assumed inviscid flow

Incompressible and Inviscid Flows• Some very important flows can be solved by neglecting

compressibility () and viscosity (away from body)• For incompressibility, the condition (derived from continuity

equation) is

• For inviscid flows (aka potential or irrigational flows), the condition is

Laplace Equation for Potential Flows• The equation is called the Laplace equation.• It is one of the most famous and extensively equations in math/physics• It has well known solutions, therefore it is easier to solve potential

(incompressible inviscid) flows analytically

• Since stream functions and potential functions are cousins, we can show for 2D flows that

Notes on the Laplace Equations• Any irrotational, incompressible flow has a velocity potential and stream

function (for 2D flow) that both satisfy Laplace’s equation.• Conversely, any solution of Laplace’s equation represents the velocity potential

or stream function (2D) for an irrotational, incompressible flow.• Laplace’s equation is a second-order linear partial differential equation.• The fact that it is linear is important, because the sum of any particular

solutions of a linear differential equation is also a solution of the equation! • Since irrotational, incompressible flow is governed by Laplace’s equation and

Laplace’s equation is linear, we conclude that a complicated flow pattern for an irrotational, incompressible flow can be synthesized by adding together a number of elementary flows that are also irrotational and incompressible.

How to solve the Laplace Equations• Our strategy is to develop flow solutions for several different

elementary flows, which by themselves may not seem to be practical flows in real life. • However, we then proceed to add (i.e., superimpose) these

elementary flows in different ways such that the resulting flow fields do pertain to practical problems.• All of these flows have the same governing equation, i.e. • How, then, do we obtain different flows for the different bodies? The

answer is found in the boundary conditions.

Boundary Conditions in Aerodynamics

• There are two boundary conditions in all external flows• Infinity boundary conditionsFar away from the body (toward infinity), in all directions, the flow approaches the uniform freestream conditions.• Wall boundary conditionsIt is impossible for the flow to penetrate the body surface.

Elementary Potential Flow 1: Uniform Flow• It is clear that

• Integrating these equations• In polar coordinates

• The flow is obviously irrotational, therefore circulation

Elementary Potential Flow 2: Source/Sink Flow

The source strength is defines as

The potentials, streams and velocities can be calculated as

Source/Sink flow obeys mass conservation everywhere except at origin since . But we accept it as a ‘singularity’

Combining Uniform Flow with Source/Sink: Rankine Halfbody

Source and Sink Pair in a Uniform Flow: Rankine Oval

Elementary Potential Flow 3: Doublet• When source and sink of equal strength are placed at the same point

• is called the doublet strength• The streamlines are families of circles with diameterWhere C is some constant

Non-lifting Flow over a cylinder

A bit more on the flow over Cylinder• We know that pressure coefficient is

• On the surface of the cylinder, we can show that

• Therefore,

• Cp varies from 1.0 at the stagnation points to −3.0 at the top/bottom.

Elementary Flow 4: Vortex Flow• All the streamlines are concentric circles about a given point.• Velocity along any given circular streamline is constant, but vary from

one streamline to another inversely with distance from the center. • Vortex flow is incompressible everywhere.• Vortex flow is irrotational everywhere except at origin, but we except

it as a singularity (exception).• To evaluate the constant, take the circulation around a given circular

streamline of radius rTherefore, for vortex flow, the circulation taken about all streamlines is the same value, namely, = −2πC.

Summary of Elementary Flows

Lifting Flow over Cylinder• If the cylinder is spinning, it will produce finite (measurable) lift.• We model this as the superimposition of uniform flow+doublet+vortex

Lift due to Spinning of Cylinder• At the surface of the cylinder (r=R), and • Since, • The drag coefficient is given by

• The lift force can be found by

The Kutta-Joukowski Theorem

• Lift per unit span is directly proportional to circulation.• This is a powerful relation in theoretical aerodynamics called the

Kutta-Joukowski theorem, obtained in ~ 1905.• Rapidly spinning cylinder can produce a much higher lift than an

airplane wing of same planform area• However, the drag on the cylinder is also much higher than a well-

designed wing. Hence, no rotating cylinders on aircrafts.• Although was derived for a circular cylinder, it applies in general to

bodies of arbitrary cross section.

K-J Theorem and the Generation of Lift

• Let curve A be any curve in the flow enclosing the airfoil. • If airfoil is producing lift, the velocity field around the airfoil will be such that

the line integral of velocity around A will be finite• The integral of velocity around any curve not including the body is zero• Hence, the lift produced by the airfoil is given by • The Kutta-Joukowski theorem states that lift per unit span on a two-

dimensional body is directly proportional to the circulation around the body.• Just like we synthesized flow over a spinning cylinder by adding a vortex to

the non-lifting flow, we can synthesize flow over an airfoil by distributing vortices all over and inside the airfoil.

Note: Lift produces circulation, not the other way round. Think why?

Numerical Source Panel Method: Our first flavor of CFD• We added elementary flows in certain ways and discovered that resulting streamlines turned

out to fit certain body shapes.• But it is not practical to randomly add elementary flows and try to match the body shape we

seek to find the flow around!• We want to specify the shape of an arbitrary body and solve for the distribution of

singularities which, in combination with a uniform stream, produce the flow over the given body.• For the moment, we will concentrate on non-lifting flows….for a reason.• This technique is called the source-panel method, a standard tool in aerodynamic industry

since the 1960s.• Numerical solution of potential flows by both source and vortex panel techniques has

revolutionized the analysis of low-speed flows. We will consider vortex methods later for lifting flows.

Recall: Source/Sink Flow

The source strength is defines as

The potentials, streams and velocities can be calculated as

Now, instead of having a single source, we want to have a number of sources placed side by side along a contour: “Source Sheet”

The Source Sheet

• Define λ = λ(s) to be source strength per unit length along s. For infinitesimal sources, ds acts like a regular line source of strength λds and depth l

• Recall that the strength of a single line source was defined as the volume flow rate per unit depth in the z direction.

• Typical units for are square meters per second, but for λ are meters per second.

λ may be negative, so it is really a combination of sources and sinks

SPM- Derivation

• Consider a point P at distance r from portion ds of the source sheet• This portion of strength produces an infinitesimally potential

• Complete velocity potential at point P, induced by the entire source sheet from a to can be found as • We can now wrap the entire body with this source sheet and

superimpose uniform flow.

SPM-Discretizing the Continuous Equations• Approximate the source sheet by n straight panels j=1,2,…,n• Let source strength per panel be constant.• The panel strengths are unknown• We want to iterate till the body surface becomes a streamline of the

flow i.e. at the surface. • ↑THIS is the boundary condition we apply at each control point (i.e.

the middle point of each panel).• Let us now put this in numbers.

SPM-Numerical TechniqueThe integration is because for a given panel ‘r’ varies as we move across the panel

The Numerical Recipe (1)• Since point P is just an arbitrary point, but we are more interested in

what is happening at the surface, let’s move P to the body surface.• Then the effect of each source panel on a given panel is

• Slope of ith panel is , but its angle with the flow is • The normal component of free stream flow to each panel is

The Numerical Recipe (2)• The normal component of velocity induced at (xi , yi ) by the source

panels is

• We can show that this can be simply expanded as

The Numerical Recipe (3)• The velocity normal to the ith panel at (xi , yi ) is

• Therefore, we seek to impose the distribution of sources which gives us zero normal velocity at each node (i.e. body becomes a streamline)

These are n linear equations with n unknowns ()

The Numerical Recipe (3)• Finally, once we find the source distribution (), we can find the actual

velocity tangent to the surface

• We can then find the pressure coefficient at each node

• The pressure forces can then be easily resolved in lift drag and moment!

Source Panel Method Applied to a Cylinder• Let us begin by dividing the

cylinder into 8 panels.• Coordinates of the ith panel’s

control point are (xi,yi), and the edge points of the panel are (Xi,Yi) and (Xi+1,Yi+1).• Let be in the x-direction.• Let us first evaluate the integrals

SPM applied to the cylinder (2)• It is easier to

deal with flow angles with tangent to the surface rather than normal to it

SPM applied to the cylinder (3)

The SPM applied to the cylinder (4)• To obtain the actual velocity over the cylinder• The velocity due to upstream flow is • Velocity induced by sources• Therefore the total velocity is• And coefficient of pressure is

SPM applied to Cylinder (5)See Matlab code developed in class