pressure update using projection method with Large-eddy simulation(LES)

sjwon1991 · February 14, 2023, 03:54

Hi,

I'd like to ask "how to update pressure using the projection method with LES"

Chorin's projection method (incompressible formulation) proposes updating pressure due to the existence of viscous term modeled by the Crank-Nicolson method as,

$G(P)=G(\phi)-\Delta t/2Re L(G(\phi))$

where P is the exact prsesure, 𝜙 is pseudo-pressure G is gradient, L is laplace operator, Re is Reynolds number and ∆𝑡 is time increment.

Since the commutation between gradient and laplace operator yields the exact pressure as,

$P=\phi-\Delta t/2Re L(\phi)$

Using LES, the viscous term containing eddy-viscosity modeled by linear-eddy viscosity model. And the pressure correction part is written as,

$\frac{\partial P}{\partial x_{i}} =\frac{\partial \phi }{\partial x_{i}}-\Delta t\frac{\partial}{\partial x_{j}}(\nu_{T}\frac{\partial^{2}\phi}{\partial x_{j}\partial x_{i}}) \\. \qquad \quad=\frac{\partial \phi }{\partial x_{i}}-\nu_{T}\Delta t\frac{\partial}{\partial x_{j}}(\frac{\partial^{2}\phi}{\partial x_{j}\partial x_{i}})-\Delta t(\frac{\partial \nu_{T}}{\partial x_{j}}\frac{\partial^{2}\phi}{\partial x_{j}\partial x_{i}})$

$\nu_{T}$ is the summation of molecular and eddy viscosity.

Since the $\nu_{T}$ is a spatial variable, I can't find a way to take $G_{i}^{-1}$ to the upper equation.

And, if I use a compressible formulation, the equation is derived as

$\frac{\partial P}{\partial x_{i}} =\frac{\partial \phi }{\partial x_{i}}-\frac{\rho\Delta t}{2}\frac{\partial}{\partial x_{j}}(\nu_{T}\frac{\partial}{\partial x_{j}}(\frac{\partial\phi}{\partial x_{i}})+\nu_{T}\frac{\partial}{\partial x_{i}}(\frac{\partial\phi}{\partial x_{j}}))$

Are there any documents to refer to for solving the above issue?

Thanks in advance

FMDenaro · February 14, 2023, 04:45

Quote:

Originally Posted by sjwon1991

Hi,

I'd like to ask "how to update pressure using the projection method with LES"

Chorin's projection method (incompressible formulation) proposes updating pressure due to the existence of viscous term modeled by the Crank-Nicolson method as,

$G(P)=G(\phi)-\Delta t/2Re L(G(\phi))$

where P is the exact prsesure, 𝜙 is pseudo-pressure G is gradient, L is laplace operator, Re is Reynolds number and ∆𝑡 is time increment.

Since the commutation between gradient and laplace operator yields the exact pressure as,

$P=\phi-\Delta t/2Re L(\phi)$

Using LES, the viscous term containing eddy-viscosity modeled by linear-eddy viscosity model. And the pressure correction part is written as,

$\frac{\partial P}{\partial x_{i}} =\frac{\partial \phi }{\partial x_{i}}-\Delta t\frac{\partial}{\partial x_{j}}(\nu_{T}\frac{\partial^{2}\phi}{\partial x_{j}\partial x_{i}}) \\. \qquad \quad=\frac{\partial \phi }{\partial x_{i}}-\nu_{T}\Delta t\frac{\partial}{\partial x_{j}}(\frac{\partial^{2}\phi}{\partial x_{j}\partial x_{i}})-\Delta t(\frac{\partial \nu_{T}}{\partial x_{j}}\frac{\partial^{2}\phi}{\partial x_{j}\partial x_{i}})$

$\nu_{T}$ is the summation of molecular and eddy viscosity.

Since the $\nu_{T}$ is a spatial variable, I can't find a way to take $G_{i}^{-1}$ to the upper equation.

And, if I use a compressible formulation, the equation is derived as

$\frac{\partial P}{\partial x_{i}} =\frac{\partial \phi }{\partial x_{i}}-\frac{\rho\Delta t}{2}\frac{\partial}{\partial x_{j}}(\nu_{T}\frac{\partial}{\partial x_{j}}(\frac{\partial\phi}{\partial x_{i}})+\nu_{T}\frac{\partial}{\partial x_{i}}(\frac{\partial\phi}{\partial x_{j}}))$

Are there any documents to refer to for solving the above issue?

Thanks in advance

First of all, why do you want to evaluate the function P? In any case, that is not a thermodynamic pressure.

Second issue, you are supposing to evaluate the SGS eddy viscosity term within the CN time integration but you should be aware that you will face the problem of a non-linear system. I would strongly discourage you to follow this strategy. This way, you will increase the complexity of the method. Use an explicit discretization such as Adams_Bashforth for the SGS terms.

Third, an issue that is not properly considered is due to the intermediate BCs for LES.

I worked some years ago on such topics, you can find details here:

https://www.researchgate.net/publica...ection_methods

https://www.researchgate.net/publica...QlHMxaCfELCqVw

sjwon1991 · February 14, 2023, 05:31

Professor, I really appreciate your reply.

1. I try to convert density with (thermodynamic + hydrodynamic) pressure to temperature by using the equation of state. As you know, P is hydrodynamic pressure. And in my simulation, I try to use a low Mach number solver using the projection method. At the early stage of my simulation, strong buoyancy force is generated due to the local hot area and it drives non-negligible hydrodynamic pressure. This is the reason why I try to calculate P.

2. Yes, you are right. The complexity of SGS eddy viscosity occurs in LHS for approximate factorization in TDMA of velocity increment for intermediate velocity. I treat those eddy viscosities as n-step data, not intermediate variables. => It would be not good for accuracy. The simulation code is designed based on hybrid RK3 method proposed by Rai and Moin. I can try to change the diffusion term using explicit RK3 method.

3. I'll check your reference.

Once again, thank you very much for your help.

Sincerely,

Won

sjwon1991 · February 14, 2023, 05:32

Professor, I really appreciate your reply.

1. I try to convert density with (thermodynamic + hydrodynamic) pressure to temperature by using the equation of state. As you know, P is hydrodynamic pressure. And in my simulation, I try to use a low Mach number solver using the projection method. At the early stage of my simulation, strong buoyancy force is generated due to the local hot area and it drives non-negligible hydrodynamic pressure. This is the reason why I try to calculate P.

2. Yes, you are right. The complexity of SGS eddy viscosity occurs in LHS for approximate factorization in TDMA of velocity increment for intermediate velocity. I treat those eddy viscosities as n-step data, not intermediate variables. => It would be not good for accuracy. The simulation code is designed based on hybrid RK3 method proposed by Rai and Moin. I can try to change the diffusion term using explicit RK3 method.

3. I'll check your reference.

Once again, thank you very much for your help.

Sincerely,

Won

FMDenaro · February 14, 2023, 05:35

Quote:

Originally Posted by sjwon1991

Professor, I really appreciate your reply.

1. I try to convert density with (thermodynamic + hydrodynamic) pressure to temperature by using the equation of state. As you know, P is hydrodynamic pressure. And in my simulation, I try to use a low Mach number solver using the projection method. At the early stage of my simulation, strong buoyancy force is generated due to the local hot area and it drives non-negligible hydrodynamic pressure. This is the reason why I try to calculate P.

2. Yes, you are right. The complexity of SGS eddy viscosity occurs in LHS for approximate factorization in TDMA of velocity increment for intermediate velocity. I treat those eddy viscosities as n-step data, not intermediate variables. => It would be not good for accuracy. The simulation code is designed based on hybrid RK3 method proposed by Rai and Moin. I can try to change the diffusion term using explicit RK3 method.

3. I'll check your reference.

Once again, thank you very much for your help.

Sincerely,

Won

Be also aware that the commutation between laplacian and divergence is not valid everywhere. For example on a wall.

lisagold661 · February 14, 2023, 09:28

The projection method is a numerical method for solving the Navier-Stokes equations, which are equations that govern fluid flow. In this method, the velocity field is updated first, and then the pressure field is updated based on the divergence of the velocity field. The steps for updating the pressure using the projection method are as follows:

Solve for the intermediate velocity field, which is obtained by applying a finite difference or finite volume method to the Navier-Stokes equations without considering the pressure term.

Compute the divergence of the intermediate velocity field, which represents the source term for the pressure update.

Solve the Poisson equation for the pressure field, which relates to the divergence of the velocity field. The Poisson equation can be solved using an iterative method or a direct method.

Update the velocity field by subtracting the gradient of the pressure field from the intermediate velocity field.

Repeat the above steps until a convergence criteria for the velocity and pressure fields is met.

February 14, 2023, 03:54	pressure update using projection method with Large-eddy simulation(LES)	#1
sjwon1991 New Member Sungjin Won Join Date: Dec 2012 Location: Seoul/Republic of Korea Posts: 15 Rep Power: 13	Hi, I'd like to ask "how to update pressure using the projection method with LES" Chorin's projection method (incompressible formulation) proposes updating pressure due to the existence of viscous term modeled by the Crank-Nicolson method as, $G(P)=G(\phi)-\Delta t/2Re L(G(\phi))$ where P is the exact prsesure, 𝜙 is pseudo-pressure G is gradient, L is laplace operator, Re is Reynolds number and ∆𝑡 is time increment. Since the commutation between gradient and laplace operator yields the exact pressure as, $P=\phi-\Delta t/2Re L(\phi)$ Using LES, the viscous term containing eddy-viscosity modeled by linear-eddy viscosity model. And the pressure correction part is written as, $\frac{\partial P}{\partial x_{i}} =\frac{\partial \phi }{\partial x_{i}}-\Delta t\frac{\partial}{\partial x_{j}}(\nu_{T}\frac{\partial^{2}\phi}{\partial x_{j}\partial x_{i}}) \\. \qquad \quad=\frac{\partial \phi }{\partial x_{i}}-\nu_{T}\Delta t\frac{\partial}{\partial x_{j}}(\frac{\partial^{2}\phi}{\partial x_{j}\partial x_{i}})-\Delta t(\frac{\partial \nu_{T}}{\partial x_{j}}\frac{\partial^{2}\phi}{\partial x_{j}\partial x_{i}})$ $\nu_{T}$ is the summation of molecular and eddy viscosity. Since the $\nu_{T}$ is a spatial variable, I can't find a way to take $G_{i}^{-1}$ to the upper equation. And, if I use a compressible formulation, the equation is derived as $\frac{\partial P}{\partial x_{i}} =\frac{\partial \phi }{\partial x_{i}}-\frac{\rho\Delta t}{2}\frac{\partial}{\partial x_{j}}(\nu_{T}\frac{\partial}{\partial x_{j}}(\frac{\partial\phi}{\partial x_{i}})+\nu_{T}\frac{\partial}{\partial x_{i}}(\frac{\partial\phi}{\partial x_{j}}))$ Are there any documents to refer to for solving the above issue? Thanks in advance

February 14, 2023, 09:28	pressure update using projection method with Large-eddy simulation(LES)	#6
lisagold661 New Member lisagold Join Date: Feb 2023 Posts: 1 Rep Power: 0	The projection method is a numerical method for solving the Navier-Stokes equations, which are equations that govern fluid flow. In this method, the velocity field is updated first, and then the pressure field is updated based on the divergence of the velocity field. The steps for updating the pressure using the projection method are as follows: Solve for the intermediate velocity field, which is obtained by applying a finite difference or finite volume method to the Navier-Stokes equations without considering the pressure term. Compute the divergence of the intermediate velocity field, which represents the source term for the pressure update. Solve the Poisson equation for the pressure field, which relates to the divergence of the velocity field. The Poisson equation can be solved using an iterative method or a direct method. Update the velocity field by subtracting the gradient of the pressure field from the intermediate velocity field. Repeat the above steps until a convergence criteria for the velocity and pressure fields is met.

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February 14, 2023, 05:31		#3
sjwon1991 New Member Sungjin Won Join Date: Dec 2012 Location: Seoul/Republic of Korea Posts: 15 Rep Power: 13	Professor, I really appreciate your reply. 1. I try to convert density with (thermodynamic + hydrodynamic) pressure to temperature by using the equation of state. As you know, P is hydrodynamic pressure. And in my simulation, I try to use a low Mach number solver using the projection method. At the early stage of my simulation, strong buoyancy force is generated due to the local hot area and it drives non-negligible hydrodynamic pressure. This is the reason why I try to calculate P. 2. Yes, you are right. The complexity of SGS eddy viscosity occurs in LHS for approximate factorization in TDMA of velocity increment for intermediate velocity. I treat those eddy viscosities as n-step data, not intermediate variables. => It would be not good for accuracy. The simulation code is designed based on hybrid RK3 method proposed by Rai and Moin. I can try to change the diffusion term using explicit RK3 method. 3. I'll check your reference. Once again, thank you very much for your help. Sincerely, Won

February 14, 2023, 05:32		#4
sjwon1991 New Member Sungjin Won Join Date: Dec 2012 Location: Seoul/Republic of Korea Posts: 15 Rep Power: 13	Professor, I really appreciate your reply. 1. I try to convert density with (thermodynamic + hydrodynamic) pressure to temperature by using the equation of state. As you know, P is hydrodynamic pressure. And in my simulation, I try to use a low Mach number solver using the projection method. At the early stage of my simulation, strong buoyancy force is generated due to the local hot area and it drives non-negligible hydrodynamic pressure. This is the reason why I try to calculate P. 2. Yes, you are right. The complexity of SGS eddy viscosity occurs in LHS for approximate factorization in TDMA of velocity increment for intermediate velocity. I treat those eddy viscosities as n-step data, not intermediate variables. => It would be not good for accuracy. The simulation code is designed based on hybrid RK3 method proposed by Rai and Moin. I can try to change the diffusion term using explicit RK3 method. 3. I'll check your reference. Once again, thank you very much for your help. Sincerely, Won