meaning of this formula

FluidKo · October 4, 2022, 04:36

Hello everybody~!
I have a question about turbulence.
I'm taking turbulence lecture but actually I have not taken Engineering mathematics 2 before.
So I have no idea of many theories those are built by Fourier.
But I should take this lecture.
https://www.youtube.com/watch?v=izdE...X2o4TX&index=6

I don't know meaning of this formula.
$\psi (x, y, t)=\phi (y)e^{i\alpha (x-ct)}$
I've read wiki about Orr–Sommerfeld equation briefly and I've seen below equation.
$u'\propto e^{i\alpha (x-ct)}$ .
I think this is equation that describes disturbance but I have no idea of meaning, especially for exponential function.

Can you let me know meaning of this formula?
Thank you!

FMDenaro · October 4, 2022, 10:21

Quote:

Originally Posted by FluidKo

Hello everybody~!
I have a question about turbulence.
I'm taking turbulence lecture but actually I have not taken Engineering mathematics 2 before.
So I have no idea of many theories those are built by Fourier.
But I should take this lecture.
https://www.youtube.com/watch?v=izdE...X2o4TX&index=6

I don't know meaning of this formula.
$\psi (x, y, t)=\phi (y)e^{i\alpha (x-ct)}$
I've read wiki about Orr–Sommerfeld equation briefly and I've seen below equation.
$u'\propto e^{i\alpha (x-ct)}$ .
I think this is equation that describes disturbance but I have no idea of meaning, especially for exponential function.

Can you let me know meaning of this formula?
Thank you!

I don't know the notes where you read this formula but:

1) if psi is a real function, in the RHS you must have a real function. Therefore phi should be complex but I don't understand why depends on y.
2) The argument (x-ct) stands for a linear wave solution with constant velocity c. That is the function simply shifts at speed c.

agd · October 4, 2022, 15:52

The function psi is a complex waveform, with the function phi being the real magnitude and the complex part representing the phase. The dependence on y for phi is typically representative of the behavior expected in flat plate boundary layers, with y being the coordinate normal to the plate. At least that's what I remember from the Orr-Sommerfeld derivations I worked through a long time ago.

LuckyTran · October 4, 2022, 16:43

$\psi (x, y, t)=\phi (y)e^{i\alpha (x-ct)}$ [/QUOTE]
means you have a parallel flow

$u'\propto e^{i\alpha (x-ct)}$
means your disturbance u' is a wave

Neither of these are really statements but rather assumptions implicit to the analysis. You are looking for wavelike solutions of parallel flows, not all possible solutions of any arbitrary flow.

mahinseo · October 5, 2022, 23:14

am writing a thesis on open channel axial water turbine simulation.

As for the setup, i am using transient, VOF (air and water) open channel, k-e RNG model, standard wall fn.

I have set up everything else normally such as the free surface level and boundary conditions such as pressure inlet, pressure outlet, and another pressure outlet at the top surface.

For geometry,
I have 2 domains, one stationary and one rotating domain for the turbine.
Both are set with numerical beach, with the stationary domain using frame motion and the rotating domain using mesh motion with the setup seen in the picture

FluidKo · October 8, 2022, 02:57

Quote:

Originally Posted by FMDenaro

I don't know the notes where you read this formula but:

1) if psi is a real function, in the RHS you must have a real function. Therefore phi should be complex but I don't understand why depends on y.
2) The argument (x-ct) stands for a linear wave solution with constant velocity c. That is the function simply shifts at speed c.

Quote:

Originally Posted by agd

The function psi is a complex waveform, with the function phi being the real magnitude and the complex part representing the phase. The dependence on y for phi is typically representative of the behavior expected in flat plate boundary layers, with y being the coordinate normal to the plate. At least that's what I remember from the Orr-Sommerfeld derivations I worked through a long time ago.

Quote:

Originally Posted by LuckyTran

$\psi (x, y, t)=\phi (y)e^{i\alpha (x-ct)}$
means you have a parallel flow

$u'\propto e^{i\alpha (x-ct)}$
means your disturbance u' is a wave

Neither of these are really statements but rather assumptions implicit to the analysis. You are looking for wavelike solutions of parallel flows, not all possible solutions of any arbitrary flow.

Aha. That is just describing fluctuation into waveform.
Thank you guys.

October 4, 2022, 04:36	meaning of this formula	#1
FluidKo Senior Member Sangho Ko Join Date: Oct 2021 Location: South Korea Posts: 135 Rep Power: 4	Hello everybody~! I have a question about turbulence. I'm taking turbulence lecture but actually I have not taken Engineering mathematics 2 before. So I have no idea of many theories those are built by Fourier. But I should take this lecture. https://www.youtube.com/watch?v=izdE...X2o4TX&index=6 I don't know meaning of this formula. $\psi (x, y, t)=\phi (y)e^{i\alpha (x-ct)}$ I've read wiki about Orr–Sommerfeld equation briefly and I've seen below equation. $u'\propto e^{i\alpha (x-ct)}$ . I think this is equation that describes disturbance but I have no idea of meaning, especially for exponential function. Can you let me know meaning of this formula? Thank you!

October 4, 2022, 15:52		#3
agd Senior Member Join Date: Jul 2009 Posts: 351 Rep Power: 18	The function psi is a complex waveform, with the function phi being the real magnitude and the complex part representing the phase. The dependence on y for phi is typically representative of the behavior expected in flat plate boundary layers, with y being the coordinate normal to the plate. At least that's what I remember from the Orr-Sommerfeld derivations I worked through a long time ago. FluidKo likes this.

October 4, 2022, 16:43		#4
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,674 Rep Power: 65	$\psi (x, y, t)=\phi (y)e^{i\alpha (x-ct)}$ [/QUOTE] means you have a parallel flow $u'\propto e^{i\alpha (x-ct)}$ means your disturbance u' is a wave Neither of these are really statements but rather assumptions implicit to the analysis. You are looking for wavelike solutions of parallel flows, not all possible solutions of any arbitrary flow. FluidKo likes this.

October 5, 2022, 23:14		#5
mahinseo Banned Join Date: Oct 2022 Posts: 1 Rep Power: 0	am writing a thesis on open channel axial water turbine simulation. As for the setup, i am using transient, VOF (air and water) open channel, k-e RNG model, standard wall fn. I have set up everything else normally such as the free surface level and boundary conditions such as pressure inlet, pressure outlet, and another pressure outlet at the top surface. For geometry, I have 2 domains, one stationary and one rotating domain for the turbine. Both are set with numerical beach, with the stationary domain using frame motion and the rotating domain using mesh motion with the setup seen in the picture FluidKo likes this.

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