incompressible flow aerodynamics
TRANSCRIPT
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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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