Interfoam: Constitutive equation and shear stresses

carboleda · July 7, 2021, 11:05

Dear foamers,

I am trying to understand some terms in the interfoam solver namely:

- fvm::laplacian(muEff, U)
- (fvc::grad(U) & fvc::grad(muEff))

To my current understanding, this terms arise from the constitutive equation of a Newtonian isotropic fluid and can be written disregarding the minus(-) sign as $\nabla \bullet T = \mu\nabla^2 U + \nabla U \bullet \nabla \mu$ or in index notation $T_{ij,j} = \mu U_{i,jj} + \mu_{,j}U_{i,j}$ where the comma (',') subscript refers to the differentiation.
The constitutive relation states that $\nabla \bullet T = \nabla \bullet (2\mu S - 2\mu (\nabla \bullet U)I/3)$
where $S = \frac{1}{2} ( \nabla U + \nabla U^T)$ .
Due to incompresibility, $\nabla \bullet U = U_{i,i} = 0$ , thus:

$\nabla \bullet T = \nabla \bullet (\mu ( \nabla U + \nabla U^T))$

In index notation:

$T_{ij,j} = (\mu (U_{i,j} + U_{j,i}))_{,j} = \mu_{,j}(U_{i,j} + U_{j,i}) + \mu (U_{i,j} + U_{j,i})_{,j} = \mu_{,j}(U_{i,j} + U_{j,i}) + \mu U_{i,jj} + \mu U_{j,ij}$
The last term $U_{j,ij}$ can be written as $(U_{j,j})_{,i}$ and $U_{j,j} = U_{i,i} = 0$ .

Leaving us with a different result than the one coded in interfoam:

$T_{ij,j} = \mu_{,j}(U_{i,j} + U_{j,i}) + \mu U_{i,jj} \neq \mu U_{i,jj} + \mu_{,j}U_{i,j}$

or in vector notation:

$\nabla \bullet T = \nabla\mu ( \nabla U + \nabla U^T) + \mu\nabla^2 U \neq \mu\nabla^2 U + \nabla U \bullet \nabla \mu$

This is the result I have so far but it is not the accepted formulation implemented in the interfoam solver, thus my questions:
Is there a mistake in my reasoning?
Have I forgotten a term or a previous step?
What are the missing steps to achieve the formulation in interfoam?

Thanks in advance for any hint you can provide me.

Carlos

carboleda · October 13, 2021, 07:50

The answer would be that:

Code:

fvm::laplacian(muEff, U)

Evaluates $(\mu U_{i,j})_{,j} = \mu_{,j}U_{i,j} + \mu U_{i,jj}$ and not only $\mu U_{i,jj}$ , as mistakenly stated in the initial post. Furthermore:

Code:

(fvc::grad(U) & fvc::grad(muEff))

Evaluates $U_{i,j} \mu_{,i} =\mu_{,j} U_{j,i}$ .
Thus:

Code:

fvm::laplacian(muEff, U)+(fvc::grad(U) & fvc::grad(muEff))

Evaluates:
$\mu_{,j}U_{i,j} + \mu U_{i,jj} + \mu_{,j} U_{j,i} = \mu_{,j}(U_{i,j}+U_{j,i}) + \mu U_{i,jj} =T_{ij,j}$

July 7, 2021, 11:05	Interfoam: Constitutive equation and shear stresses	#1
carboleda New Member Carlos Arboleda Join Date: Jan 2016 Posts: 2 Rep Power: 0	Dear foamers, I am trying to understand some terms in the interfoam solver namely: - fvm::laplacian(muEff, U) - (fvc::grad(U) & fvc::grad(muEff)) To my current understanding, this terms arise from the constitutive equation of a Newtonian isotropic fluid and can be written disregarding the minus(-) sign as $\nabla \bullet T = \mu\nabla^2 U + \nabla U \bullet \nabla \mu$ or in index notation $T_{ij,j} = \mu U_{i,jj} + \mu_{,j}U_{i,j}$ where the comma (',') subscript refers to the differentiation. The constitutive relation states that $\nabla \bullet T = \nabla \bullet (2\mu S - 2\mu (\nabla \bullet U)I/3)$ where $S = \frac{1}{2} ( \nabla U + \nabla U^T)$ . Due to incompresibility, $\nabla \bullet U = U_{i,i} = 0$ , thus: $\nabla \bullet T = \nabla \bullet (\mu ( \nabla U + \nabla U^T))$ In index notation: $T_{ij,j} = (\mu (U_{i,j} + U_{j,i}))_{,j} = \mu_{,j}(U_{i,j} + U_{j,i}) + \mu (U_{i,j} + U_{j,i})_{,j} = \mu_{,j}(U_{i,j} + U_{j,i}) + \mu U_{i,jj} + \mu U_{j,ij}$ The last term $U_{j,ij}$ can be written as $(U_{j,j})_{,i}$ and $U_{j,j} = U_{i,i} = 0$ . Leaving us with a different result than the one coded in interfoam: $T_{ij,j} = \mu_{,j}(U_{i,j} + U_{j,i}) + \mu U_{i,jj} \neq \mu U_{i,jj} + \mu_{,j}U_{i,j}$ or in vector notation: $\nabla \bullet T = \nabla\mu ( \nabla U + \nabla U^T) + \mu\nabla^2 U \neq \mu\nabla^2 U + \nabla U \bullet \nabla \mu$ This is the result I have so far but it is not the accepted formulation implemented in the interfoam solver, thus my questions: Is there a mistake in my reasoning? Have I forgotten a term or a previous step? What are the missing steps to achieve the formulation in interfoam? Thanks in advance for any hint you can provide me. Carlos Last edited by carboleda; July 12, 2021 at 06:50.

October 13, 2021, 07:50	Tentative solution	#2
carboleda New Member Carlos Arboleda Join Date: Jan 2016 Posts: 2 Rep Power: 0	The answer would be that: Code: fvm::laplacian(muEff, U) Evaluates $(\mu U_{i,j})_{,j} = \mu_{,j}U_{i,j} + \mu U_{i,jj}$ and not only $\mu U_{i,jj}$ , as mistakenly stated in the initial post. Furthermore: Code: (fvc::grad(U) & fvc::grad(muEff)) Evaluates $U_{i,j} \mu_{,i} =\mu_{,j} U_{j,i}$ . Thus: Code: fvm::laplacian(muEff, U)+(fvc::grad(U) & fvc::grad(muEff)) Evaluates: $\mu_{,j}U_{i,j} + \mu U_{i,jj} + \mu_{,j} U_{j,i} = \mu_{,j}(U_{i,j}+U_{j,i}) + \mu U_{i,jj} =T_{ij,j}$

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