Incompressible Flow Aerodynamics

8/13/2019 Incompressible Flow Aerodynamics

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Incompressible flow

AerodynamicsTheory, formulas and problems

ISA, basics on l ift and drag, continuity, momentum andenergy equation, streamline, stream function, velocity

potential, basic flows, symmetric thin airfoil theory, lifting

line theory.

Ramanathan V

Dept. of Aeronautics, Anna university

Dept. of Physics, Bharathiyar University

India.


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FormulasProperties of Atmosphere

T = T0h, - lapse rate, h- height above the sea level

Determination of Drag and Lift over bodies

Forces on the body are due to two parameters

1. Shear force over the surface2. Pressure distribution normal to the surface

On the upper surface, on the lower surface,


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In terms of non dimensional co efficient, the lift and drag

Example :


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SPECIAL CASES

CONE AT HYPERSONIC FLOW

We cant use the same formula since the cone is a 3 dimensional body, so from the

figure,


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Force on the surface of the cone is

CENTRE OF PRESSURE:

When the aerodynamic forces are to be represented in terms of N and A then to

represent their position on the chord we specify a point called cop, where the moment produced by the

N and A is same as that is produced by the body due to the distributed loads,

Alternative way of expressing the moment when the force and moment on some other point is known :

this is generally preffered bcoz the former method cop goes to

infinity when N is zero.


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In terms of nondimensional coefficient:

Flow Similarity:

Since the coeff icients depend on Reynolds numbers and Mach number for flow to be

similar the flow must have the parameters identical.

General formulas:

Level flight

Fluid Statics:


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Hence,

Vector notations and some important definitions

Gradient

Physically gradient is the representation of the rate of change of the quantity in the

specified direction.

Divergence of a vector field

Physically means the time rate of change of the volume of moving fluid element with

fixed mass per unit volume.


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Curl

Physically represents the half of angular velocity of the element under consideration,

it measures sine component of the element under consideration.

Theorem connecting the line, surface and volume integral:

Mass flux:


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Drag force when the velocity profile @ inlet and outlet are given:

Substantial Derivative: ----- average time rate of range of the property

Streamline:

Streamlines are lines whose tangent at any point gives the direction of velocity vector

at that point.

So,


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Angular velocity, Vorticity, Strain:

Angular velocity about z axis:

Vorticity:

General method of representation of the angular velocity is through Vorticity, which

is numerically twice the magnitude of the angular velocity and is quite useful since it appears more

generally in equations.

Strain:

Refers to the change in the angle, positive strain corresponds to decreasing angle.

Change in angle from the figure,

Strain,


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Time rate of strain:

Matrix of components:

Circulation:

Circulation is the negative line integral of velocity along the contour.

Positive circulation is clockwise whereas the line integral is positive in counterclockwise direction hence

reasoning the negative sign in the equation.

Stream Function:


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Stream fn in polar coordinates:

Incompressible equations

Velocity Potential:

Relation b/w velocity potential and stream function:


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Basic Flow Equations:


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Euler equation and Bernoulli equation:

Rotational flow

Irrotational flow

Pressure

Coefficient:

Condition for incompressibility: Condition for irrotationality: (angular moment

about the z axis is zero for 2d flows)

Laplace Equationfor both incompressible and Irrotational flows only

Laplace equation in terms of stream function:


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Boundary Conditions:

Infinity Boundary condition:

Far away from the object flow approaches uniform free stream condition in all

directions, where only u is present and v is zero.

Wall Boundary Condition:

At the wall the normal velocity to the wall is zero, only the velocity component in

tangential direction is present. (Inviscid, incompressible, Irrotational flow)

Flow tangency condition for flow where velocity components are known.

Procedure:


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FUNDAMENTAL FLOWvelocity potential, stream function, radial velocity, angular velocity:

Uniform Flow

Source and Sink flow


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Combination of Uniform flow with source and sink flow:

Stagnation point : (r,o) =

Combination of Uniform, Source and Sink (equal strength separated by distance 2b):

Stagnation point from origin,


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Doublet Flow:

From the equation:

From analytical geometry it represents the equation of circle with d as the diameter.


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Non lifting flow over a cylinder: (combination of uniform and doublet flow)

STAGNATION POINTS


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Flow can be replaced with a circular cylinder of radius R and a freestream flow with velocity V.


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Vortex Flow:

LIFTING FLOW OVER THE CIRCULAR CYLINDER:


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Incompressible Flow Aerodynamics