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In my Fluid Mechanics class, we arrived at an equation for 2D compressible flow (using perturbation theory):

$$\left(1-M^{2}\right)\frac{\partial^{2}\phi}{\partial x^{2}}+\frac{\partial^{2}\phi}{\partial y^{2}}=0$$

where $\phi$ is the perturbed fluid potential ($\vec v=\vec U+\vec\nabla(\phi)$), $\vec U=U\vec e_{x}$ is a constant flow and $M=U/c_s$ is the Mach number.

I wanted to resolve this equation for an arbitrary contour defined by $$ f_\pm=\begin{cases}f(x) & \quad\text{if }x\in[0,l] \\ 0 & \quad \text{if } x\in]l,+\infty[\end{cases} $$

where $f_{+}$ is for $y>0$ and $f_{-}$ for $y<0$. and you also asume $\frac{df}{dx}\ll(\text{the other non pertubed quantities})$

My teacher only solved the supersonic case using the fact that for $M>1$ you get the wave equation. He found his initial conditions like this:

$$\frac{df}{dx}=\tan(\alpha)=\frac{\nabla\phi|_{y}}{U+\nabla\phi|_{y}}$$

wich gives neglecting the quadratic perturbed quantities:

$$\frac{\partial\phi}{\partial y}|_{y=f'(x)}\simeq\frac{\partial\phi}{\partial y}_{y=0}\simeq Uf'(x).$$

My problem is that I have tried solving this equation (using Fourier transform) for a subsonic flow using the same initial condition (and a bounded solution). And while I found a solution, this solution doesn't seem very physical or accurate when I plot it on python.

So I was wondering if my Limit conditions are incorrect and if they are what are the correct one?

In my Fluid Mechanics class, we arrived at an equation for 2D compressible flow (using perturbation theory):

$$\left(1-M^{2}\right)\frac{\partial^{2}\phi}{\partial x^{2}}+\frac{\partial^{2}\phi}{\partial y^{2}}=0$$

where $\phi$ is the perturbed fluid potential ($\vec v=\vec U+\vec\nabla(\phi)$), $\vec U=U\vec e_{x}$ is a constant flow and $M=U/c_s$ is the Mach number.

I wanted to resolve this equation for an arbitrary contour defined by $$ f_\pm=\begin{cases}f(x) & \quad\text{if }x\in[0,l] \\ 0 & \quad \text{if } x\in]l,+\infty[\end{cases} $$

where $f_{+}$ is for $y>0$ and $f_{-}$ for $y<0$. and you also asume $\frac{df}{dx}\ll(\text{the other non pertubed quantities})$

My teacher only solved the supersonic case using the fact that for $M>1$ you get the wave equation. He found his boundary conditions like this:

$$\frac{df}{dx}=\tan(\alpha)=\frac{\nabla\phi|_{y}}{U+\nabla\phi|_{y}}$$

wich gives neglecting the quadratic perturbed quantities:

$$\frac{\partial\phi}{\partial y}|_{y=f'(x)}\simeq\frac{\partial\phi}{\partial y}_{y=0}\simeq Uf'(x).$$

My problem is that I have tried solving this equation (using Fourier transform) for a subsonic flow using the same boundary condition (and a bounded solution). And while I found a solution, this solution doesn't seem very physical or accurate when I plot it on python.

So I was wondering if my boundary conditions are incorrect and if they are what are the correct one?

in my Fluid Mechanics class, we arrived at an equation for 2D compressible flow (using perturbation theory):

$(1-M^{2})\frac{\partial^{2}\phi}{\partial x^{2}}+\frac{\partial^{2}\phi}{\partial y^{2}}$=0

where $\phi$ is the perturbed fluid potential ($\vec v=\vec U+\vec\nabla(\phi)$), $\vec U=U\vec e_{x}$ is a constant flow and $M=\frac{U}{c_{s}}$ is the Mach number.

I wanted to resolve this equation for an arbitrary contour defined by $f_{\pm}(x)=\begin{array}{ll}f(x)\, if\,\,x\in[0,l]\\0\,if\,\,x\in\,]l,+\infty[ \end{array}$

where $f_{+}$ is for $y>0$ and $f_{-}$ for $y<0$. and you also asume $\frac{df}{dx}<<(the\,other\, non\, pertubed\, quantities)$

my teacher only solved the supersonic case using the fact that for $M>1$ you get the wave equation. He found his initial conditions like this:

$\frac{df}{dx}=tan(\alpha)=\frac{\nabla\phi|_{y}}{U+\nabla\phi|_{y}}$

wich gives neglecting the quadratic perturbed quantities:

$\frac{\partial\phi}{\partial y}|_{y=f'(x)}\simeq\frac{\partial\phi}{\partial y}_{y=0}\simeq Uf'(x)$.

My problem is that I have tried solving this equation (using Fourier transform) for a subsonic flow using the same initial condition (and a bounded solution). And while I found a solution, this solution doesn't seem very physical or accurate when I plot it on python.

So I was wondering if my Limit conditions are incorrect and if they are what are the correct one?

In my Fluid Mechanics class, we arrived at an equation for 2D compressible flow (using perturbation theory):

$$\left(1-M^{2}\right)\frac{\partial^{2}\phi}{\partial x^{2}}+\frac{\partial^{2}\phi}{\partial y^{2}}=0$$

where $\phi$ is the perturbed fluid potential ($\vec v=\vec U+\vec\nabla(\phi)$), $\vec U=U\vec e_{x}$ is a constant flow and $M=U/c_s$ is the Mach number.

I wanted to resolve this equation for an arbitrary contour defined by $$ f_\pm=\begin{cases}f(x) & \quad\text{if }x\in[0,l] \\ 0 & \quad \text{if } x\in]l,+\infty[\end{cases} $$

where $f_{+}$ is for $y>0$ and $f_{-}$ for $y<0$. and you also asume $\frac{df}{dx}\ll(\text{the other non pertubed quantities})$

My teacher only solved the supersonic case using the fact that for $M>1$ you get the wave equation. He found his initial conditions like this:

$$\frac{df}{dx}=\tan(\alpha)=\frac{\nabla\phi|_{y}}{U+\nabla\phi|_{y}}$$

wich gives neglecting the quadratic perturbed quantities:

$$\frac{\partial\phi}{\partial y}|_{y=f'(x)}\simeq\frac{\partial\phi}{\partial y}_{y=0}\simeq Uf'(x).$$

My problem is that I have tried solving this equation (using Fourier transform) for a subsonic flow using the same initial condition (and a bounded solution). And while I found a solution, this solution doesn't seem very physical or accurate when I plot it on python.

So I was wondering if my Limit conditions are incorrect and if they are what are the correct one?

in my Fluid Mechanics class, we arrived at an equation for 2D compressible flow (using perturbation theory):

$(1-M^{2})\frac{\partial^{2}\phi}{\partial x^{2}}+\frac{\partial^{2}\phi}{\partial y^{2}}$=0

where $\phi$ is the perturbed fluid potential ($\vec v=\vec U+\vec\nabla(\phi)$), $\vec U=U\vec e_{x}$ is a constant flow and $M=\frac{U}{c_{s}}$ is the Mach number.

I wanted to resolve this equation for an arbitrary contour defined by $f_{\pm}(x)=\begin{array}{ll}f(x)\, if\,\,x\in[0,l]\\0\,if\,\,x\in\,]l,+\infty[ \end{array}$

where $f_{+}$ is for $y>0$ and $f_{-}$ for $y<0$. and you also asume $\frac{df}{dx}<<(the\,other\, non\, pertubed\, quantities)$

my teacher only solved the supersonic case using the fact that for $M>1$ you get the wave equation. He found his initial conditions like this:

$\frac{df}{dx}=tan(\alpha)=\frac{\nabla\phi|_{y}}{U+\nabla\phi|_{y}}$

wich gives neglecting the quadratic perturbed quantities:

$\frac{\partial\phi}{\partial y}|_{y=f'(x)}\simeq\frac{\partial\phi}{\partial y}_{y=0}\simeq Uf'(x)$.

My problem is that I have tried solving this equation (using Fourier transform) for a subsonic flow using the same initial condition (and a bounded solution). And while I found a solution, this solution doesn't seem very physical or accurate when I plot it on python.

in my Fluid Mechanics class, we arrived at an equation for 2D compressible flow (using perturbation theory):

$(1-M^{2})\frac{\partial^{2}\phi}{\partial x^{2}}+\frac{\partial^{2}\phi}{\partial y^{2}}$=0

where $\phi$ is the perturbed fluid potential ($\vec v=\vec U+\vec\nabla(\phi)$), $\vec U=U\vec e_{x}$ is a constant flow and $M=\frac{U}{c_{s}}$ is the Mach number.

I wanted to resolve this equation for an arbitrary contour defined by $f_{\pm}(x)=\begin{array}{ll}f(x)\, if\,\,x\in[0,l]\\0\,if\,\,x\in\,]l,+\infty[ \end{array}$

where $f_{+}$ is for $y>0$ and $f_{-}$ for $y<0$. and you also asume $\frac{df}{dx}<<(the\,other\, non\, pertubed\, quantities)$

my teacher only solved the supersonic case using the fact that for $M>1$ you get the wave equation. He found his initial conditions like this:

$\frac{df}{dx}=tan(\alpha)=\frac{\nabla\phi|_{y}}{U+\nabla\phi|_{y}}$

wich gives neglecting the quadratic perturbed quantities:

$\frac{\partial\phi}{\partial y}|_{y=f'(x)}\simeq\frac{\partial\phi}{\partial y}_{y=0}\simeq Uf'(x)$.

My problem is that I have tried solving this equation (using Fourier transform) for a subsonic flow using the same initial condition (and a bounded solution). And while I found a solution, this solution doesn't seem very physical or accurate when I plot it on python.

So I was wondering if my Limit conditions are incorrect and if they are what are the correct one?