Question on streamlines around a Rankine Half Body

[influx]

I have several questions that I am seeking clarification on. The first 2 are rather simple:

1) The formula for r describes the blue streamline (the stagnation streamline) right? (Just confirming)
2) I assume that the stagnation streamline is called so because it passes through the stagnation point?
3) What is the difference between Ψ = +πm and πa ? They both seem to refer to the same thing? I mean the distance from the y=0 to the top of the stagnation stream line is equal to both +πm and πa ?
4) Why does rsin(theta) = πa ? I know that rsin(theta) = y and y = πa but what is the significance of labelling the vertical distance as rsin(theta)? Any specific reason behind this?

 

[Chestermiller]

There is a limiting streamline that encloses the flow eminating from the inner line source. The stream function is equal to πm on this limiting streamline (meaning that the enclosed volumetric flow rate per unit width is equal to πm between y = 0 and the limiting streamline). The contour of the limiting streamline becomes independent of x at large values of x (small θ), and the y location of the streamline in this asymptote is πa.

 

[influx]

There is a limiting streamline that encloses the flow eminating from the inner line source. The stream function is equal to πm on this limiting streamline (meaning that the enclosed volumetric flow rate per unit width is equal to πm between y = 0 and the limiting streamline). The contour of the limiting streamline becomes independent of x at large values of x (small θ), and the y location of the streamline in this asymptote is πa.

So the limiting stream line (the stagnation stream line) has a stream function = πm anywhere along the line? Meaning the value of the stream function doesn't change anywhere along the blue line? That does make sense since the stream function is constant along any stream line but then what confuses me is the formula for a Rankine Half Body:

Ψ(x,y) = U_∞y + m(arctan(y/x))

According to the above, the stream function is a function of x and y so surely the stream function should vary along the blue streamline (according to the formula)?

 

[Chestermiller]

So the limiting stream line (the stagnation stream line) has a stream function = πm anywhere along the line?

Yes.

Meaning the value of the stream function doesn't change anywhere along the blue line?

Yes.

That does make sense since the stream function is constant along any stream line but then what confuses me is the formula for a Rankine Half Body:

Ψ(x,y) = U_∞y + m(arctan(y/x))

According to the above, the stream function is a function of x and y so surely the stream function should vary along the blue streamline (according to the formula)?

For very large values of x (compared to y), the second term vanishes, and the stream function equation approaches

Ψ(x,y) = U_∞y

For the streamline Ψ=πm, this gives the asymptote of the blue streamline as
$$y=\frac{πm}{U_∞}=πa$$
where $a=m/U_∞$

So, at large values of x, all the streamlines become parallel to the x axis, and the stream function becomes a function only of y. In this large x region, the velocity becomes horizontal and has a uniform value of just U_∞.

 

[influx]

Yes.

Yes.

For very large values of x (compared to y), the second term vanishes, and the stream function equation approaches

Ψ(x,y) = U_∞y

For the streamline Ψ=πm, this gives the asymptote of the blue streamline as
$$y=\frac{πm}{U_∞}=πa$$
where $a=m/U_∞$

So, at large values of x, all the streamlines become parallel to the x axis, and the stream function becomes a function only of y. In this large x region, the velocity becomes horizontal and has a uniform value of just U_∞.

So at large x values, the second term disappears. Understood. So what about at small x values? For small x-values (compared to y) the second term does not disappear hence the stream function is still reliant upon x and consequently the stream function cannot be constant for the entire stream line? It can only be constant for the section of the stream line where the second term becomes negligible?

For example, for the section of the stagnation stream line circled in purple below:

The stagnation streamline hasn't become parallel to the x-axis yet meaning that it is still a function of x. So how is the stream function at any point within this purple circle equal to the stream function outside of it?

Thanks

 

[Chestermiller]

So at large x values, the second term disappears. Understood. So what about at small x values? For small x-values (compared to y) the second term does not disappear hence the stream function is still reliant upon x and consequently the stream function cannot be constant for the entire stream line?

Yes, it is.
If we set Ψ = mπ, the equation for the blue streamline becomes:
$$U_∞y+m\tan^{-1}(y/x)=mπ\tag{1}$$. If you plot y vs x for this equation, you will get the blue streamline.

It can only be constant for the section of the stream line where the second term becomes negligible?

No.

For example, for the section of the stagnation stream line circled in purple below:

The stagnation streamline hasn't become parallel to the x-axis yet meaning that it is still a function of x. So how is the stream function at any point within this purple circle equal to the stream function outside of it?

If you substitute y = r sin θ and x = r cos θ into Eqn. 1 for the blue streamline, what do you get?

Chet

 

[influx]

Yes, it is.
If we set Ψ = mπ, the equation for the blue streamline becomes:
$$U_∞y+m\tan^{-1}(y/x)=mπ\tag{1}$$. If you plot y vs x for this equation, you will get the blue streamline.

No.

If you substitute y = r sin θ and x = r cos θ into Eqn. 1 for the blue streamline, what do you get?

Chet

Thanks for the reply and apologies for my late reply.

If we substitute y = r sin θ and x = r cos θ into Eqn. 1 for the blue streamline we yield:

U_∞(rsinθ) + m(arctan(tan(θ))) = mπ

Unsure of how to progress further..

 

[Chestermiller]

Thanks for the reply and apologies for my late reply.

If we substitute y = r sin θ and x = r cos θ into Eqn. 1 for the blue streamline we yield:

U_∞(rsinθ) + m(arctan(tan(θ))) = mπ

Unsure of how to progress further..

arctan(tanθ) = θ, correct? For arctan, we substitute the words "the angle whose tangent is."