Aerodynamics HW - fluid particles acceleration based on stream functio

The stream function is defined as ψ=Axy, where A is a constant and x and y represent the coordinates of a particle. Using the given equations for velocity and acceleration, Chetan calculates the components of velocity and acceleration, but is unable to match the given equation of acceleration. They are unsure where the factor of r and (x2-y2) come from, and seek further help. In summary, Chetan is attempting to prove an equation for the acceleration of particles in a corner flow using a given stream function, but is unable to match the given equation due to unknown factors.
  • #1
leonida
10
0

Homework Statement


prove that with flow in a corner, with stream function ψ=Axy, particles are accelerating per [itex]\frac{DV}{Dt}[/itex]=(A2(x2-y2))/r; A=const; r-distance from the center of the corner


Homework Equations



Vx=U=[itex]\frac{∂ψ}{∂y}[/itex] . . Vy=V=-[itex]\frac{∂ψ}{∂x}[/itex]

a=[itex]\frac{∂V}{∂t}[/itex]+U[itex]\frac{∂V}{∂x}[/itex]+[itex]\frac{∂V}{∂y}[/itex]

The Attempt at a Solution



As per above equations i get velocity components as
U=Ax and V=-Ay


then since local acc is 0 acceleration is:

a=Ax[itex]\frac{A(x-y)}{∂x}[/itex] - Ay[itex]\frac{A(x-y)}{∂y}[/itex]

finally, as per my calcs, accelerations is:

a=A2(x+y)

where did this r come from and also (x2-y2). i was thinking using r2=x2+y2, and using to multiply the whole acceleration expression with r2/(x2+y2), but i am getting nowhere.

help please
 
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  • #2
Your equation does not give the acceleration. The acceleration is a vector, and you need to find its components first, before determining its magnitude.

[tex]a_x=u\frac{∂u}{∂x}+v\frac{∂u}{∂y}[/tex]
[tex]a_y=u\frac{∂v}{∂x}+v\frac{∂v}{∂y}[/tex]

Even with this, you still don't match the answer you are trying to prove.

Chet
 

Related to Aerodynamics HW - fluid particles acceleration based on stream functio

1. What is aerodynamics?

Aerodynamics is the study of how air and other gases interact with objects in motion, particularly in relation to the forces of lift and drag.

2. What is a stream function?

A stream function is a mathematical function used in fluid dynamics to represent the motion of fluid particles in a two-dimensional flow field. It is a useful tool for visualizing and analyzing fluid flow patterns.

3. How is fluid particle acceleration calculated based on the stream function?

Fluid particle acceleration can be calculated using the Euler equations, which are a set of equations that describe the motion of an ideal fluid. These equations use the stream function to calculate the velocity and acceleration of fluid particles at any given point in the flow field.

4. What factors affect fluid particle acceleration in aerodynamics?

The acceleration of fluid particles in aerodynamics is affected by a number of factors, including the shape and size of the object, the viscosity of the fluid, the speed of the object, and the angle of attack (the angle between the object's direction of motion and the direction of the air flow).

5. How is the study of aerodynamics important in real-world applications?

Aerodynamics has numerous practical applications, including designing aircraft, cars, and other vehicles, optimizing wind turbine efficiency, and understanding weather patterns. It is also crucial in sports such as cycling, skiing, and bobsledding, where minimizing drag can significantly improve performance.

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