Problems implementing axisymmetric Method of Characteristics

pedrolcn · June 3, 2016, 17:35

Hi

I'm currently trying to implement the method of characteristics for axisymmetric irrotational compressible flow as part of a project of implementing Rao's method for the design of optimum thrust nozzle contours, but I've ran into some issues I hope you can shed some light on.

We have the following governing equations:

$(u^{2}-a^{2})u_{x}+(v^{2}-a^{2})v_{y}+2uvu_{y}-\frac{a^{2}v}{y}$
$u_{y}-v_{x}=0$

Which yields the following characteristic equation:

$(\frac{dy}{dx})_{\stackrel{+}{-}}=(\lambda)_{\stackrel{+}{-}}=tan(\theta\stackrel{+}{-}\alpha)$

And the following compatibility relation which is valid along the mach lines:

$(u^{2}-a^{2})du_{\stackrel{+}{-}}+[2uv-(u^{2}-a^{2})\lambda_{\stackrel{+}{-}}]dv_{\stackrel{+}{-}}-(a^{2}v/y)dx_{\stackrel{+}{-}}$

My issue is when implementing the unit process for an interior point, if the point from which the left-running characteristic originates sits on the axis of symmetry, the term $(a^{2}v/y)$ goes to infinity, so how should I treat this equation, should I consider that the whole term $(a^{2}v/y)$ goes to zero, since on the symmetry axis we also have

?

Thank you, Pedro N

usv001 · July 31, 2016, 01:43

Hi Pedro,

I am new to MOC myself but I have developed my own code for planar MOC successfully and I am trying to develop my own code for axisymmetric MOC as well. I just thought we could help each other.

To answer your question: No, the term

does not go to 0 in the limit of $y\rightarrow0$ . It approaches $\left(a^2V\frac{d\theta}{dy}\right)_{y=0} \ne 0$ . So, if you have a point very close to the axis where you know $V, \theta$ and

, you can probably approximate the term by $\frac{a^2V\theta}{y}$ .

I am facing a similar problem actually. I am working with the Mach number

and the flow angle $\theta$ as my variables as opposed to

and

. Given the throat height (or mass flow rate) and design Mach number, how do you set up the initial conditions to kick-start the solution process? Unlike the planar case, the Riemann invariants $K_{\pm} = \theta \mp \nu$ do not remain constant along the characteristics. So, how do you set up the starting characteristic when you have no characteristics to intersect it?

Many thanks.

June 3, 2016, 17:35	Problems implementing axisymmetric Method of Characteristics	#1
pedrolcn New Member Pedro Nascimento Join Date: Jun 2016 Posts: 1 Rep Power: 0	Hi I'm currently trying to implement the method of characteristics for axisymmetric irrotational compressible flow as part of a project of implementing Rao's method for the design of optimum thrust nozzle contours, but I've ran into some issues I hope you can shed some light on. We have the following governing equations: $(u^{2}-a^{2})u_{x}+(v^{2}-a^{2})v_{y}+2uvu_{y}-\frac{a^{2}v}{y}$ $u_{y}-v_{x}=0$ Which yields the following characteristic equation: $(\frac{dy}{dx})_{\stackrel{+}{-}}=(\lambda)_{\stackrel{+}{-}}=tan(\theta\stackrel{+}{-}\alpha)$ And the following compatibility relation which is valid along the mach lines: $(u^{2}-a^{2})du_{\stackrel{+}{-}}+[2uv-(u^{2}-a^{2})\lambda_{\stackrel{+}{-}}]dv_{\stackrel{+}{-}}-(a^{2}v/y)dx_{\stackrel{+}{-}}$ My issue is when implementing the unit process for an interior point, if the point from which the left-running characteristic originates sits on the axis of symmetry, the term $(a^{2}v/y)$ goes to infinity, so how should I treat this equation, should I consider that the whole term $(a^{2}v/y)$ goes to zero, since on the symmetry axis we also have ? Thank you, Pedro N

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July 31, 2016, 01:43		#2
usv001 Senior Member Join Date: Sep 2015 Location: Singapore Posts: 102 Rep Power: 10	Hi Pedro, I am new to MOC myself but I have developed my own code for planar MOC successfully and I am trying to develop my own code for axisymmetric MOC as well. I just thought we could help each other. To answer your question: No, the term does not go to 0 in the limit of $y\rightarrow0$ . It approaches $\left(a^2V\frac{d\theta}{dy}\right)_{y=0} \ne 0$ . So, if you have a point very close to the axis where you know $V, \theta$ and , you can probably approximate the term by $\frac{a^2V\theta}{y}$ . I am facing a similar problem actually. I am working with the Mach number and the flow angle $\theta$ as my variables as opposed to and . Given the throat height (or mass flow rate) and design Mach number, how do you set up the initial conditions to kick-start the solution process? Unlike the planar case, the Riemann invariants $K_{\pm} = \theta \mp \nu$ do not remain constant along the characteristics. So, how do you set up the starting characteristic when you have no characteristics to intersect it? Many thanks.