Asking help on derivation of ALE differential form of momentum conservation equation

pfguo · April 26, 2024, 05:02

I want to find the differential equation form of momentum conservation equation that holds at any point
within an arbitrary moving and deforming grid, i.e., the momentum differential equation in ALE description.

After reading the wonderful book "fluid mechanics" sixth edition by PIJUSH K. KUNDU et. al, i.e., equation 4.17,
if I understan right, the ALE form of Reynolds transport is

$\frac{D}{D t} \int_{V^*(t)} \rho \mathbf{u} d V^*=\frac{\partial}{\partial t} \int_{V^*(t)} \rho \mathbf{u} d V^*+\underbrace{\int_{A^*(t)} \rho \mathbf{u}(\mathbf{u}-\mathbf{b}) \cdot d \mathbf{A}^*}_{\text {Term } *}$

Let handle the volume intergral in the above equation as

$\frac{\partial}{\partial t} \int_{V^*(t)} \rho \mathbf{u} d V^*=\int_{V^*(t)} \frac{\partial(\rho \mathbf{u})}{\partial t} d V^*+\underbrace{\int_{A^*(t)} \rho \mathbf{u} \mathbf{b} \cdot d \mathbf{A}^*}_{\text {Term }{ }^{**}}$

You see that when I moved the differentiation into the integral,
a surface integral term (Term**) containing the boundary velocity appeared.

This term would cancel out the term containing the surface velocity ( $\textbf{b}$ ) in the Reynolds transport equation,
ultimately leading to the absence of surface velocity in the transport equation.

I believe this is an incorrect result.

Can you give me some suggestions on where I am doing wrong?

Thanks!

Sincerely,

Guo Pengfei

agd · April 29, 2024, 15:50

Try using Gauss' theorem to convert the area integrals to volume integrals. Then you have to make the argument that if the integral equals to zero then under sufficient conditions for continuity of the integrand that the integrand must equal zero over the domain. Since these conditions fail in the presence of discontinuities most people use the integral form, or figure out the singular limit conditions to apply to the jumps where the differential form is invalid.

agd · April 29, 2024, 15:53

You will also need to move the temporal derivative inside the volume integral, even in the case of deforming boundaries. There is a mathematical theorem to handle that also - I'll leave that as an exercise for the reader.

pfguo · April 30, 2024, 05:56

Quote:

Originally Posted by agd

You will also need to move the temporal derivative inside the volume integral, even in the case of deforming boundaries. There is a mathematical theorem to handle that also - I'll leave that as an exercise for the reader.

I really appreciate for your attention.

Actually, I got stuck even before applying the divergence theorem to the surface integral term.

I got stuck on the term involving the derivative of the volume integral with respect to time.

When I moved the differentiation inside the integral, a surface integral term (Term**) containing the boundary velocity appeared.

This term would cancel out the term containing the surface velocity ( $\textbf{b}$ ) in the Reynolds transport equation,
ultimately leading to the absence of surface velocity in the transport equation.

I believe this is an incorrect result.

By the way, for better readability, I spent some time replacing the original image with LaTeX formulas

agd · April 30, 2024, 10:48

Are you sure you aren't applying Liebniz's theorem twice? Consider mass conservation for a fluid element - mass conservation in a Lagrangian sense is simply the time derivative of the integral of density over the volume is zero. Then you can apply Liebniz's theorem to get the equation where the temporal derivative is inside the integral and the relative velocity appears in the area integral. Then apply divergence to that. It seems like you are double-counting the surface velocity.

FMDenaro · April 30, 2024, 10:55

Just follow the idea of the derivation of the Reynolds transport theorem.
You can read a rigorous description in the textbook of Chorin & Marsden.

pfguo · May 1, 2024, 00:01

Quote:

Originally Posted by agd

Are you sure you aren't applying Liebniz's theorem twice? Consider mass conservation for a fluid element - mass conservation in a Lagrangian sense is simply the time derivative of the integral of density over the volume is zero. Then you can apply Liebniz's theorem to get the equation where the temporal derivative is inside the integral and the relative velocity appears in the area integral. Then apply divergence to that. It seems like you are double-counting the surface velocity.

Currently, the part where I'm stuck doesn't involve Gauss's theorem, and I don't think my problem lies in the use of Leibniz's theorem either. I strongly feel that I may have misunderstood the quantity whether in Euler's description or or ALE's description and their derivatives with respect to space or time.

pfguo · May 1, 2024, 00:03

Quote:

Originally Posted by FMDenaro

Just follow the idea of the derivation of the Reynolds transport theorem.
You can read a rigorous description in the textbook of Chorin & Marsden.

Thank you for recommending the book. I've been reading it for a while, but I'm still unclear about where my problem lies. Nevertheless, I sincerely appreciate your attention.

FMDenaro · May 1, 2024, 12:34

Quote:

Originally Posted by pfguo

Thank you for recommending the book. I've been reading it for a while, but I'm still unclear about where my problem lies. Nevertheless, I sincerely appreciate your attention.

Follow carefully the derivation of Chorin

@ss · May 4, 2024, 04:09

Hello,
I am working on the FSI problem for vortex-induced vibration. The ALE approach has been used to solve coupling motion for fluid and solid. The fluid equation is discretized based on FVM, and the structure equation is the newmark method used, and that is FEM. However, I am not sure about how the mesh motion equation, which is a Laplacian equation, is solved to update mesh motion velocity. and also, don't you know if this method is used for it FVM or FEM? Please help me to clear this up and also suggest some books that can give a good understanding of the ALE approach computationally. I am using OpenFOAM for my computations.

Thank you

pfguo · May 5, 2024, 21:34

Quote:

Originally Posted by FMDenaro

Follow carefully the derivation of Chorin

Thank you for your suggestion.

I have been reading the content you mentioned in the book you recommended these past few days.

Basically, I can understand the proofs in the book, but it doesn't relate directly to the question I asked.

I kindly request you to read my question again.

April 26, 2024, 05:02	Derivation of ALE differential form of momentum conservation equation	#1
pfguo New Member pengfeiguo Join Date: Nov 2017 Posts: 10 Rep Power: 8	I want to find the differential equation form of momentum conservation equation that holds at any point within an arbitrary moving and deforming grid, i.e., the momentum differential equation in ALE description. After reading the wonderful book "fluid mechanics" sixth edition by PIJUSH K. KUNDU et. al, i.e., equation 4.17, if I understan right, the ALE form of Reynolds transport is $\frac{D}{D t} \int_{V^(t)} \rho \mathbf{u} d V^=\frac{\partial}{\partial t} \int_{V^(t)} \rho \mathbf{u} d V^+\underbrace{\int_{A^(t)} \rho \mathbf{u}(\mathbf{u}-\mathbf{b}) \cdot d \mathbf{A}^}_{\text {Term } }$ Let handle the volume intergral in the above equation as $\frac{\partial}{\partial t} \int_{V^(t)} \rho \mathbf{u} d V^=\int_{V^(t)} \frac{\partial(\rho \mathbf{u})}{\partial t} d V^+\underbrace{\int_{A^(t)} \rho \mathbf{u} \mathbf{b} \cdot d \mathbf{A}^}_{\text {Term }{ }^{}}$ You see that when I moved the differentiation into the integral, a surface integral term (Term) containing the boundary velocity appeared. This term would cancel out the term containing the surface velocity ( $\textbf{b}$ ) in the Reynolds transport equation, ultimately leading to the absence of surface velocity in the transport equation. I believe this is an incorrect result. Can you give me some suggestions on where I am doing wrong? Thanks! Sincerely, Guo Pengfei Last edited by pfguo; April 30, 2024 at 05:51. Reason: edit for more clear question*

May 4, 2024, 04:09	How the lapacian eqation for mesh motion is solved in openfoam	#10
@ss New Member shristi Join Date: Jun 2023 Posts: 26 Rep Power: 3	Hello, I am working on the FSI problem for vortex-induced vibration. The ALE approach has been used to solve coupling motion for fluid and solid. The fluid equation is discretized based on FVM, and the structure equation is the newmark method used, and that is FEM. However, I am not sure about how the mesh motion equation, which is a Laplacian equation, is solved to update mesh motion velocity. and also, don't you know if this method is used for it FVM or FEM? Please help me to clear this up and also suggest some books that can give a good understanding of the ALE approach computationally. I am using OpenFOAM for my computations. Thank you

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April 29, 2024, 15:50		#2
agd Senior Member Join Date: Jul 2009 Posts: 357 Rep Power: 19	Try using Gauss' theorem to convert the area integrals to volume integrals. Then you have to make the argument that if the integral equals to zero then under sufficient conditions for continuity of the integrand that the integrand must equal zero over the domain. Since these conditions fail in the presence of discontinuities most people use the integral form, or figure out the singular limit conditions to apply to the jumps where the differential form is invalid.

April 29, 2024, 15:53		#3
agd Senior Member Join Date: Jul 2009 Posts: 357 Rep Power: 19	You will also need to move the temporal derivative inside the volume integral, even in the case of deforming boundaries. There is a mathematical theorem to handle that also - I'll leave that as an exercise for the reader.

April 30, 2024, 10:48		#5
agd Senior Member Join Date: Jul 2009 Posts: 357 Rep Power: 19	Are you sure you aren't applying Liebniz's theorem twice? Consider mass conservation for a fluid element - mass conservation in a Lagrangian sense is simply the time derivative of the integral of density over the volume is zero. Then you can apply Liebniz's theorem to get the equation where the temporal derivative is inside the integral and the relative velocity appears in the area integral. Then apply divergence to that. It seems like you are double-counting the surface velocity.

April 30, 2024, 10:55		#6
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,849 Rep Power: 73	Just follow the idea of the derivation of the Reynolds transport theorem. You can read a rigorous description in the textbook of Chorin & Marsden.