| Computational Aerodynamics Questions & Answers | |
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$\tau$ is the same as $t$ because we set $\Gamma$ to 1. 1 point bonus.
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No, $\xi$ and $\eta$ don't have dimensions. But their derivatives do of course. So $\xi_x$, $\eta_x$, etc will have dimensions (1/m or 1/mm). 1 point bonus.
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Try again now.
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Consider a system of equations as follows: $$ \frac{\partial U}{\partial t}+ A \frac{\partial U}{\partial x}=0 $$ Recall that the eigenvalues are such that $A=L^{-1}\Lambda L$: $$ \frac{\partial U}{\partial t} + L^{-1} \Lambda L \frac{\partial U}{\partial x}=0 $$ Multiply by $L$: $$ L\frac{\partial U}{\partial t} + \Lambda L \frac{\partial U}{\partial x}=0 $$ Say that a vector $W$ exists such that $L=\partial W/\partial U$. Further, because $L=L(U)$, it follows that $W=W(U)$. Thus, we can say $$ \frac{\partial W}{\partial t}=L\frac{\partial U}{\partial t} ~~{\rm and}~~ \frac{\partial W}{\partial x}=L\frac{\partial U}{\partial x} $$ Substitute the RHS of the latter 2 equations in the former: $$ \frac{\partial W}{\partial t} + \Lambda \frac{\partial W}{\partial x}=0 $$ Because $\Lambda=[\phi_1,~\phi_2,~\phi_3]^{\rm D}$ is diagonal, the latter is simply a list of advection equations each with a wave speed $\phi_1$, $\phi_2$, $\phi_3$. Good question: 2 points bonus.
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The problem with your logic is that $\partial \xi_x / \partial \xi$ is not zero because $\xi_x$ can vary along the the $\xi$ coordinate. Similarly, $\partial \eta_x / \partial \eta$ is also not zero because $\eta_x$ can vary along $\eta$. Good question, 2 points bonus.
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The problem in your math is here: $$ \overset{\sim}{W_0}+\overset{\sim}{W_1}=\frac{1}{(\epsilon + \beta_0)^2+(\epsilon + \beta_1)^2} $$ This is not correct. Rather: \begin{align} \overset{\sim}{W_0}+\overset{\sim}{W_1}&=\frac{\gamma_0}{(\epsilon + \beta_0)^2}+\frac{\gamma_1}{(\epsilon + \beta_1)^2}\\ &=\frac{\gamma_0 (\epsilon + \beta_1)^2+ \gamma_1 (\epsilon + \beta_0)^2}{(\epsilon + \beta_0)^2(\epsilon + \beta_1)^2}\\ \end{align}
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The cowl starts where the domain ends. So, there is no need to grid the cowl and there's hence no need to know the cowl length. The important thing is that your waves (either the Mach waves for the Prandtl-Meyer compression fan case or the shock waves for the 3-oblique-shock case) all meet at the same point and such a point is located exactly at the domain exit.
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Yes, for problem 3, there should be 3 oblique shocks and the pressure ratio across each shock should be the same. You need to design your inlet so that these 3 shocks appear, have all the same pressure ratio, and meet at one point. I am not sure what you don't understand. Can you explain better the problem?
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Your grid has issues. You should make the grid so that the spacing is more or less uniform everywhere. Don't worry about making the grid parallel to the shocks. The shocks can go through the cells at an angle — this is fine. Simply make sure that your bottom wall has the right shape, that the cells are more or less of the same size everywhere (and are not distorted as in your mesh), and the shocks will appear correctly positioned.
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The instructions are in the CFDWARP HOWTO here: |
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Please attach a picture of the pressure contours around your cylinder. Maybe your flow is not yet at steady-state. Because you are doing the cylinder case, you don't need the xstation. Inside the Post() module, delete the part about the x-station and only keep the line Fpressure[dim]=_Fpressure(... Then, add a bit of code to find the drag coefficient using Fpressure[1] (the pressure force acting on the body along $x$).
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Problem 4 is harder than the others. You need to find $\sigma$ and $\delta$ for each shock using the obliqueshock program. Once this is done, you need to find the bottom surface of the ramp that will make all shocks meet each other at one point (such is only function of $H$, and the 3 $\delta$s and $\sigma$s). Thus, this requires a bit of geometrical calculations on paper. I recommend another problem if this is taking too long.
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| $\pi$ |